49:
1786:. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years. Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe. In 1884 a young German mathematician,
2003:
3425:
3669:
7342:, then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic. (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.
726:
12040:
121:
3191:
2992:
3420:{\displaystyle H=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{matrix}}\right):\,\theta \in \mathbb {R} \right\}\subset \mathbb {T} ^{2}=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right):\,\theta ,\phi \in \mathbb {R} \right\},}
10715:
are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might
8182:
The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map
1541:
along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains
10439:
Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Like solvable groups, nilpotent groups are
6632:
The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not even obvious that the Lie algebra is independent of
4164:
Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall, Rossmann, and
Stillwell. Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential
1561:. Rotating a circle is an example of a continuous symmetry. For any rotation of the circle, there exists the same symmetry, and concatenation of such rotations makes them into the circle group, an archetypal example of a Lie group. Lie groups are widely used in many parts of modern mathematics and
10313:
Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object
5228:
is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the
1781:
considered the winter of 1873â1874 as the birth date of his theory of continuous groups. Thomas
Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation. Some of Lie's early
6564:
6295:
To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are
10435:
Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Solvable groups are too messy to classify except in a few small
1985:
Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly
6907:
if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the
10074:
One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through
2664:
9265:
2847:
1930:
Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by
3169:
10687:
in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not
5604:
In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. There are (up to isomorphism) only two Lie algebras of dimension two. The associated simply connected Lie groups are
7296:, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.
10678:
Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to model them locally on
6760:
Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element
10484:. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group
6450:
7700:
says that every finite-dimensional real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a
8032:
5420:
The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent,
10637:
This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.
10296:
One can also study (in general infinite-dimensional) unitary representations of an arbitrary Lie group (not necessarily compact). For example, it is possible to give a relatively simple explicit description of the
1869:. However, the hope that Lie theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a
7197:
5294:
10337:
ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the
9460:
7559:
1990:(i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups. The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by
3464:
2340:
9572:
5219:
5165:
5121:
5061:
4880:
7299:
On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the
2987:{\displaystyle SO(2,\mathbb {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\,\varphi \in \mathbb {R} /2\pi \mathbb {Z} \right\}.}
10696:
topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold.
2537:
9069:
8847:
8117:
7925:
7883:
1888:, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory:
1670:
1630:
7838:
5938:
8302:
9893:
9933:
9585:
9379:
4739:
10240:
10118:
6993:. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a
3084:
9490:
7611:
2526:
2418:
10366:, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E
7285:
Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras.
7074:
4814:
4668:
4448:
4295:
4255:
4158:
4066:
3038:
2804:
2764:
2485:
8805:
8470:
6008:
4628:
4408:
2140:
Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system. Here, the
6831:
6238:
5632:
3834:
3719:
3663:
3503:
2703:
2111:
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8624:
8593:
8228:
7519:
7495:
6889:
6855:
5722:
2161:
9413:
8764:
8738:
8342:
7222:
from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity.
6104:
5487:
4936:
asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952,
4906:
4517:
4365:
4343:
4215:
4193:
4114:
1744:
1722:
1171:
1146:
1109:
7644:
5970:
10151:
9602:, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of
7671:
7144:
7101:
5671:
5599:
5325:
4774:
4694:
4552:
4474:
4321:
4092:
3073:
1801:
Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of
1698:
9061:
6044:
2832:
10290:
7760:
5394:
5361:
3951:
10716:
not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.
6671:, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
6626:
3745:
3075:
real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form
10195:
10171:
7691:
6799:
at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of
6131:
5769:
5544:
5517:
4584:
3911:
3884:
5573:
10260:
10048:
10028:
10008:
9977:
9957:
9844:
9824:
9801:
9777:
9738:
9718:
9694:
9670:
9650:
9308:
9288:
9005:
8974:
8954:
8934:
8914:
8890:
8870:
8716:
8696:
8644:
8533:
8513:
8493:
8402:
8382:
8362:
8268:
8248:
8204:
8177:
8157:
8137:
8075:
8055:
7579:
7471:
7451:
7424:
7404:
7378:
7275:
7255:
6196:
6176:
5804:
5742:
5691:
4014:
3991:
3971:
3931:
3857:
3788:
3765:
3690:
3630:
3610:
3590:
3570:
3550:
3530:
1590:
10446:
are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example,
1821:
had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for
7292:, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use
10699:
The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix
6559:{\displaystyle \operatorname {Lie} (G)=\{X\in M(n;\mathbb {C} )|\operatorname {exp} (tX)\in G{\text{ for all }}t{\text{ in }}\mathbb {\mathbb {R} } \},}
6158:
Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector space, or the space of smooth functions from a manifold
10263:
9987:
7697:
5634:(with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples".
7335:. These two groups have isomorphic Lie algebras, but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.
6150:
Lie groups (except in the trivial sense that any group having at most countably many elements can be viewed as a 0-dimensional Lie group, with the
538:
6392:, where Δ is an infinitesimal positive number with Δ = 0 (of course, no such real number Δ exists). For example, the orthogonal group O(
12899:
7937:
12090:
1473:
10464:
Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras. They are central extensions of products of simple Lie groups.
2165:
586:
11817:
10788:
There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have
9612:
is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on
10030:. Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of
6918:
using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on
12894:
7040:
The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a
2183:, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a
11266:
7152:
591:
6633:
the representation we use. To get around these problems we give the general definition of the Lie algebra of a Lie group (in 4 steps):
12181:
10416:, which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup.
5015:
that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an
2149:
1790:, came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume
581:
576:
2210:
realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of
2072:
that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties
12205:
10286:
10065:
12400:
8984:
7351:
6922:
can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space
6290:
5248:
5238:
1979:
396:
9418:
7524:
11791:
10277:(including the just-mentioned case of SO(3)) is particularly tractable. In that case, every finite-dimensional representation of
660:
543:
10325:
The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a
10270:, essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation.
7033:. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real)
10744:
1031:
17:
12270:
12002:
11933:
11906:
11876:
11849:
11753:
11731:
11702:
11617:
11570:
11532:
11486:
11425:
11393:
11362:
10819:
6804:
3433:
10727:
of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the
1813:. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany. Lie's
12496:
6259:-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "
2279:
981:
691:
9498:
12549:
12077:
5184:
5130:
5086:
5026:
4845:
3040:, and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.
10281:
decomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified by
2659:{\displaystyle GL(2,\mathbb {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\,\det A=ad-bc\neq 0\right\}.}
2157:
12833:
11971:
11644:
9260:{\displaystyle \exp(X)\,\exp(Y)=\exp \left(X+Y+{\tfrac {1}{2}}+{\tfrac {1}{12}},Y]-{\tfrac {1}{12}},X]-\cdots \right),}
1466:
976:
11775:
11667:
11471:
11463:
11455:
7788:
6648:
of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket =
1511:
92:
70:
10748:
4964:), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many
63:
12598:
8810:
8080:
7888:
7846:
5338:
be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group
1635:
1595:
7804:
5901:
2053:
or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its
12581:
12190:
10342:
of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A
10298:
8273:
4020:" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.
1787:
553:
10318:
himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have
9864:
4952:
is a topological manifold with continuous group operations, then there exists exactly one analytic structure on
1750:, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician
11542:
9898:
3164:{\displaystyle A=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quad a>0,\,b\in \mathbb {R} .}
1866:
1392:
10740:
9313:
12793:
12200:
11554:
11417:
10844:
10839:
10059:
7709:
7328:
6642:
5644:
4699:
2141:
1806:
1459:
548:
528:
10204:
10082:
7044:. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let
2262:
1829:
tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of
1546:
where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are
12778:
12501:
12275:
9983:
9613:
7798:
7782:
7237:
which is also a group homomorphism. Observe that, by the above, a continuous homomorphism from a Lie group
6657:
4957:
2806:. It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is
493:
401:
9465:
7584:
7426:. This notion is important because of the following result that has simple connectedness as a hypothesis:
6320:(In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.)
2490:
2372:
12823:
12044:
6133:. In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying
4987:
of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to
1076:
890:
10289:
is in terms of the "highest weight" of the representation. The classification is closely related to the
7047:
4883:
4779:
4633:
4413:
4260:
4220:
4123:
4031:
3632:
winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a
3003:
2769:
2729:
2450:
12828:
12798:
12506:
12462:
12443:
12210:
12154:
11601:
11546:
10302:
10069:
8773:
8410:
7769:
Methods for determining whether a Lie group is simply connected or not are discussed in the article on
7104:
6248:
5981:
4933:
4596:
4376:
3510:
2245:
Lie groups play an important role, via their connections with Galois representations in number theory.
2172:
2145:
1975:
1952:
1870:
808:
533:
11380:
BĂ€uerle, G. G. A.; de Kerf, E. A.; ten Kroode, A. P. E. (1997). A. van
Groesen; E.M. de Jager (eds.).
10707:. On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix
10177:, which has a single spherical orbital.) This assumption does not necessarily mean that the solutions
7728:
in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group
6809:
6201:
5608:
3793:
3695:
3639:
3479:
2679:
2218:. This insight opened new possibilities in pure algebra, by providing a uniform construction for most
2087:
1919:
12365:
12230:
10590:
10455:
9010:
8657:
8605:
8545:
8209:
7500:
7476:
6870:
6836:
6431:
The preceding description can be made more rigorous as follows. The Lie algebra of a closed subgroup
5438:
7717:
5696:
1817:
was to develop a theory of symmetries of differential equations that would accomplish for them what
12750:
12615:
12307:
12149:
10834:
10766:
10488:
can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write
7203:
5404:
acquires a structure of a manifold near the identity element. One then shows that the group law on
1874:
1274:
1008:
885:
773:
684:
168:
57:
9386:
8747:
8721:
7720:.) The failure of SO(3) to be simply connected is intimately connected to the distinction between
6304:
of two such infinitesimal elements. Before giving the abstract definition we give a few examples:
6087:
5470:
4889:
4479:
4348:
4326:
4198:
4176:
4097:
1727:
1705:
1154:
1129:
1092:
12447:
12417:
12341:
12331:
12287:
12117:
12070:
10262:). This space, therefore, constitutes a representation of SO(3). These representations have been
9752:
9744:
8718:. This exponential map is a generalization of the exponential function for real numbers (because
8651:
7616:
6067:
5943:
5368:
4117:
3046:
2073:
2034:
1758:. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of
1523:
11918:
Continuous
Symmetries, Lie algebras, Differential Equations and Computer Algebra: second edition
10454:
is simple according to the second definition but not according to the first. They have all been
10127:
10076:
7770:
7355:
7233:
homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a
5519:
of complex numbers with absolute value one (with the group operation being multiplication). The
4991:. This categorical point of view leads also to a different generalization of Lie groups, namely
1904:
12788:
12407:
12302:
12215:
12122:
12020:
10755:
10402:
9847:
9592:
7649:
7382:
7289:
7122:
7079:
5650:
5578:
5397:
5242:
4984:
4744:
4673:
4522:
4453:
4300:
4071:
3052:
2719:
1948:
1826:
1759:
1677:
1424:
1214:
488:
451:
419:
406:
74:
11273:
10761:
The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a
9034:
8179:
that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.
6013:
3933:
is identified homeomorphically with the real line by identifying each element with the number
2817:
12437:
12432:
11683:, Sources and Studies in the History of Mathematics and Physical Sciences, Berlin, New York:
10782:
10732:
10326:
9270:
where the omitted terms are known and involve Lie brackets of four or more elements. In case
8766:
is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for
8741:
8307:
7763:
7739:
5840:
5833:
4926:-adic manifold, such that the group operations are analytic. In particular, each point has a
3936:
2192:
2038:
1900:
1850:
1298:
520:
188:
10424:
are all known: they are finite central quotients of a product of copies of the circle group
9621:, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.
6581:
3724:
263:
253:
243:
233:
12768:
12706:
12554:
12258:
12248:
12220:
12195:
12105:
12012:
11943:
11886:
11712:
11580:
11435:
11382:
11372:
10770:
10693:
10180:
10156:
10121:
8536:
7841:
7713:
7676:
7332:
6324:
6109:
5747:
5522:
5495:
4988:
4557:
4170:
3889:
3862:
2835:
2709:; it has two connected components corresponding to the positive and negative values of the
2444:
2191:
on the manifold. Linear actions of Lie groups are especially important, and are studied in
1960:
1755:
1673:
1569:
1238:
1226:
844:
778:
148:
138:
5549:
5299:
3185:
number of elements that is not a Lie group under a certain topology. The group given by
1873:, but it was developed by others, such as Picard and Vessiot, and it provides a theory of
8:
12942:
12906:
12588:
12466:
12451:
12380:
12139:
11763:
11598:
Lie groups, physics, and geometry: an introduction for physicists, engineers and chemists
10793:
10654:
10330:
9599:
9578:
7324:
5434:
4969:
2258:
2219:
2215:
2180:
2176:
2122:
1893:
1878:
1858:
1810:
1554:
1519:
813:
708:
677:
665:
506:
336:
12879:
11786:
11678:
5374:
5341:
12947:
12848:
12803:
12700:
12571:
12375:
12063:
11524:
10849:
10469:
10413:
10267:
10245:
10033:
10013:
9993:
9962:
9942:
9936:
9829:
9809:
9786:
9762:
9748:
9723:
9703:
9679:
9655:
9635:
9293:
9273:
8990:
8959:
8939:
8919:
8899:
8875:
8855:
8701:
8681:
8629:
8518:
8498:
8478:
8387:
8367:
8347:
8253:
8233:
8189:
8162:
8142:
8122:
8060:
8040:
7928:
7792:
7564:
7456:
7436:
7409:
7389:
7363:
7260:
7240:
7026:
6571:
6181:
6161:
5789:
5727:
5676:
3999:
3976:
3956:
3916:
3842:
3773:
3750:
3675:
3615:
3595:
3575:
3555:
3535:
3515:
2223:
2114:
2077:
1802:
1767:
1575:
798:
770:
437:
427:
12385:
10633:/1 is nilpotent, and therefore its ascending central series has all quotients abelian.
6709:. This shows that the space of left invariant vector fields (vector fields satisfying
3790:
can, however, be given a different topology, in which the distance between two points
2350:
1818:
12952:
12783:
12763:
12758:
12665:
12576:
12390:
12370:
12225:
12164:
11998:
11967:
11929:
11902:
11872:
11845:
11771:
11749:
11727:
11698:
11663:
11640:
11613:
11584:
11566:
11528:
11504:
11482:
11467:
11459:
11451:
11421:
11389:
11358:
10829:
10610:
10421:
7725:
6994:
6982:
6151:
5807:
5012:
4965:
4834:
3506:
3470:
2440:
2153:
2042:
1203:
1046:
940:
501:
464:
33:
11831:
11808:
5829:
4960:). If the underlying manifold is allowed to be infinite-dimensional (for example, a
1915:
1846:
1369:
616:
354:
12921:
12715:
12670:
12593:
12564:
12422:
12355:
12350:
12345:
12335:
12127:
12110:
11990:
11959:
11921:
11864:
11826:
11804:
11741:
11688:
11632:
11605:
11558:
11442:
11350:
11267:"Introduction to Lie groups and algebras : Definitions, examples and problems"
10986:
10876:
10774:
10736:
10461:
10443:
10378:
10339:
10334:
7339:
7293:
5896:
5889:
5880:
5871:
5862:
5853:
5844:
5779:
5456:
5452:
4961:
4941:
4839:
4590:
2366:. The two requirements can be combined to the single requirement that the mapping
2227:
2207:
2184:
2126:
2069:
1991:
1885:
1822:
1543:
1538:
1497:
1354:
1346:
1338:
1330:
1322:
1310:
1250:
1190:
1180:
1022:
964:
839:
636:
316:
308:
300:
292:
284:
217:
198:
158:
11116:
9618:
3099:
27:
Group that is also a differentiable manifold with group operations that are smooth
12864:
12773:
12603:
12559:
12325:
12008:
11986:
11939:
11882:
11708:
11684:
11576:
11516:
11431:
11411:
11368:
10689:
10684:
10586:
10473:
10398:
10386:
7215:
7041:
7030:
6997:, and it is equal to twice the one defined through left-invariant vector fields.
6300:
close" to the identity, and the Lie bracket of the Lie algebra is related to the
6081:
5974:
5822:
5783:
5772:
5448:
5008:
4996:
4976:
4821:
3182:
2706:
2211:
2188:
2118:
2081:
1932:
1892:
The idea of symmetry, as exemplified by Galois through the algebraic notion of a
1747:
1534:
1438:
1431:
1417:
1374:
1262:
1185:
1015:
929:
869:
749:
621:
374:
359:
130:
12730:
12655:
12625:
12523:
12516:
12456:
12427:
12297:
12292:
12253:
11951:
11659:
10724:
10662:
10650:
10643:
10481:
10477:
10429:
10382:
8647:
7303:
structure of a Lie group is not determined by its Lie algebra; for example, if
7234:
6264:
6137:
cover, one guarantees a group structure (compatible with its other structures).
4937:
3993:
is just the group of real numbers under addition and is therefore a Lie group.
2266:
2011:
2007:
1964:
1834:
1798:
first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse.
1445:
1381:
1071:
1051:
988:
953:
874:
864:
849:
834:
788:
765:
641:
459:
364:
11994:
11963:
11868:
11693:
11636:
11562:
1944:
1861:, the driving conception was of a theory capable of unifying, by the study of
1526:, such that group multiplication and taking inverses are both differentiable.
626:
12936:
12916:
12740:
12735:
12720:
12710:
12660:
12637:
12511:
12471:
12412:
12360:
12159:
11861:
Lie groups and algebras with applications to physics, geometry, and mechanics
11748:, Progress in Mathematics, vol. 140 (2nd ed.), Boston: BirkhÀuser,
11609:
11588:
11399:
11384:
Finite and infinite dimensional Lie algebras and their application in physics
10852:, about the application of Lie groups to the study of differential equations.
10666:
10406:
10394:
10390:
10319:
10174:
9697:
9595:
from the functor Lie to the identity functor on the category of Lie groups.)
8767:
7116:
7029:. In the case of complex Lie groups, such a homomorphism is required to be a
6789:
6297:
6275:
can be Lie groups (of course they must also have a differentiable structure).
5814:
5174:
4916:
4370:
3668:
2807:
2673:
2239:
2231:
2199:
2019:
1971:
1854:
1763:
1364:
1286:
1120:
993:
859:
349:
178:
12050:
1926:
and others, and culminated in
Riemann's revolutionary vision of the subject.
12843:
12838:
12680:
12647:
12620:
12528:
12169:
11985:, Graduate Texts in Mathematics, vol. 94, New York Berlin Heidelberg:
11407:
10824:
10728:
10680:
10658:
10618:
10374:
10282:
10066:
Compact group § Representation theory of a connected compact Lie group
9859:
8270:
at the identity), one proves that there is a unique one-parameter subgroup
8159:; thus, we have an exponential map for all matrix groups. Every element of
8027:{\displaystyle \exp(X)=1+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\cdots }
7721:
7034:
6252:
5888:
have dimensions 14, 52, 78, 133, and 248. Along with the A-B-C-D series of
5490:
5444:
5409:
4992:
4980:
2203:
2023:
1956:
1838:
1558:
1219:
918:
907:
854:
829:
824:
783:
754:
717:
646:
631:
432:
414:
344:
11508:
10975:"Sur les invariants différentiels des groupes continus de transformations"
12686:
12675:
12632:
12533:
12134:
11894:
11631:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
11629:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
10762:
7211:
6757:) on a Lie group is a Lie algebra under the Lie bracket of vector fields.
2710:
2437:
2130:
2065:
2054:
1842:
1783:
1487:
472:
388:
112:
11503:. Cambridge Tracts in Mathematical Physics. Cambridge University Press.
9803:âi.e. a Lie subgroup such that the inclusion map is a smooth embedding.
9598:
The exponential map from the Lie algebra to the Lie group is not always
3000:
Addition of the angles corresponds to multiplication of the elements of
2187:
on a manifold places strong constraints on its geometry and facilitates
12911:
12869:
12695:
12608:
12240:
12144:
12055:
11716:
10991:
10778:
10754:
The diffeomorphism group of spacetime sometimes appears in attempts to
10315:
7207:
6301:
6263:-adic Lie groups".) In general, only topological groups having similar
6077:
The quotient of a Lie group by a closed normal subgroup is a Lie group.
6063:
4945:
2270:
2235:
2031:
1778:
1751:
1547:
1386:
1114:
611:
477:
369:
11899:
Lie
Algebras and Lie Groups: 1964 Lectures given at Harvard University
11117:"Lectures on Lie Groups and Representations of Locally Compact Groups"
10909:
8515:. The operation on the right hand side is the group multiplication in
6055:
There are several standard ways to form new Lie groups from old ones:
5327:
share the same Lie algebra; thus, they are locally isomorphic. Hence,
2002:
12725:
12690:
12282:
11557:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
11496:
10888:
Hall only claims smoothness, but the same argument shows analyticity.
10747:. Diffeomorphism groups of compact manifolds of larger dimension are
9780:
9741:
7230:
3633:
2045:. One of the key ideas in the theory of Lie groups is to replace the
1923:
37:
11357:, Chicago Lectures in Mathematics, Chicago: Univ. of Chicago Press,
10974:
10781:, and have central extensions whose Lie algebras are (more or less)
9584:
7708:
An example of a simply connected group is the special unitary group
6891:
is finite-dimensional and it has the same dimension as the manifold
6765:
of the tangent space at the identity is the vector field defined by
2152:. Groups whose representations are of particular importance include
1884:
Additional impetus to consider continuous groups came from ideas of
1849:. The initial application that Lie had in mind was to the theory of
12889:
12884:
12874:
12265:
12086:
11272:. State University of New York at Stony Brook. 2006. Archived from
10797:
10373:
Lie groups are classified according to their algebraic properties (
10075:
representation theory. Consider, for example, the time-independent
10050:
which determines which subalgebras correspond to closed subgroups.
9756:
8535:. The formal similarity of this formula with the one valid for the
7581:
is simply connected, then there is a unique Lie group homomorphism
7192:{\displaystyle \phi _{*}\colon {\mathfrak {g}}\to {\mathfrak {h}},}
6578:
is a real vector space that is closed under the bracket operation,
5818:
5467:
The only connected Lie groups with dimension one are the real line
2723:
2061:
1911:
1862:
1530:
744:
11416:, History of Mathematics, vol. 21, Providence, Rhode Island:
7327:
for examples). An example of importance in physics are the groups
6255:
of an infinite extension of fields, or the additive group of the
5459:), and these give most of the more common examples of Lie groups.
5443:
Lie groups occur in abundance throughout mathematics and physics.
5425:
together with the group law determines the geometry of the group.
1754:(1842â1899), who laid the foundations of the theory of continuous
11925:
11844:, Oxford Graduate Texts in Mathematics, Oxford University Press,
10692:. Instead one needs to define Lie groups modeled on more general
9806:
Examples of non-closed subgroups are plentiful; for example take
9383:
The exponential map relates Lie group homomorphisms. That is, if
7733:
7219:
1937:
1562:
1086:
1000:
568:
10197:
are rotationally invariant functions. Rather, it means that the
9462:
the induced map on the corresponding Lie algebras, then for all
8852:
Because the exponential map is surjective on some neighbourhood
8807:
with the regular commutator is the Lie algebra of the Lie group
7277:
is an isomorphism of Lie groups if and only if it is bijective.
6084:
of a connected Lie group is a Lie group. For example, the group
5892:, the exceptional groups complete the list of simple Lie groups.
4999:
in the category of smooth manifolds with a further requirement.
12481:
12039:
11388:. Studies in mathematical physics. Vol. 7. North-Holland.
10447:
9673:
9604:
5221:; that is, a matrix Lie group satisfies the above conditions.)
4979:
provides a concise definition for Lie groups: a Lie group is a
4165:
map. The following are standard examples of matrix Lie groups.
2811:
2015:
725:
9310:
commute, this formula reduces to the familiar exponential law
6284:
2084:
of distance-preserving transformations of the
Euclidean space
1503:
10428:
and simple compact Lie groups (which correspond to connected
10291:
classification of representations of a semisimple Lie algebra
9990:
between the connected Lie subgroups of a connected Lie group
7729:
6066:
subgroup of a Lie group is a Lie group. This is known as the
5289:{\displaystyle G'\subset \operatorname {GL} (n,\mathbb {C} )}
3474:
1935:, who in 1888 published the first paper in a series entitled
9455:{\displaystyle \phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}}
7554:{\displaystyle f:{\mathfrak {g}}\rightarrow {\mathfrak {h}}}
7280:
6408: = 1, so the Lie algebra consists of the matrices
4820:
All of the preceding examples fall under the heading of the
11855:. The 2003 reprint corrects several typographical mistakes.
6939:
can also be described as follows: the commutator operation
6903:
up to "local isomorphism", where two Lie groups are called
120:
11901:, Lecture notes in mathematics, vol. 1500, Springer,
10743:
for a derivation of this fact) is the symmetry algebra of
1941:
The composition of continuous finite transformation groups
6674:
We apply this construction to the case when the manifold
1943:). The work of Killing, later refined and generalized by
12021:"Lie Groups. Representation Theory and Symmetric Spaces"
11413:
Essays in the history of Lie groups and algebraic groups
11379:
11333:
2125:
one is interested in the properties invariant under the
11724:
Differential
Geometry, Lie Groups, and Symmetric Spaces
11315:
10440:
too messy to classify except in a few small dimensions.
10153:
commutes with the action of SO(3) on the wave function
7338:
On the other hand, if we require that the Lie group be
4882:), and holomorphic maps. Similarly, using an alternate
3459:{\displaystyle a\in \mathbb {R} \setminus \mathbb {Q} }
11983:
Foundations of differentiable manifolds and Lie groups
10932:
10930:
10242:
is invariant under rotations (for each fixed value of
9204:
9161:
9131:
8979:
The exponential map and the Lie algebra determine the
3325:
3211:
2887:
2583:
2234:, deals extensively with analogues of Lie groups over
1762:, in much the same way that finite groups are used in
1553:
Lie groups provide a natural model for the concept of
11011:
10804:) gauge theory becomes an 11-dimensional theory when
10719:
Some of the examples that have been studied include:
10248:
10207:
10183:
10159:
10130:
10124:
as a symmetry, meaning that the Hamiltonian operator
10085:
10036:
10016:
9996:
9965:
9945:
9901:
9867:
9832:
9812:
9789:
9765:
9726:
9706:
9682:
9658:
9638:
9501:
9468:
9421:
9389:
9316:
9296:
9276:
9072:
9037:
9013:
8993:
8962:
8942:
8922:
8902:
8878:
8858:
8813:
8776:
8750:
8724:
8704:
8684:
8660:
8632:
8608:
8548:
8521:
8501:
8481:
8413:
8390:
8370:
8350:
8310:
8276:
8256:
8236:
8212:
8192:
8165:
8145:
8125:
8083:
8063:
8043:
7940:
7891:
7849:
7807:
7742:
7679:
7652:
7619:
7587:
7567:
7527:
7503:
7479:
7459:
7439:
7412:
7392:
7366:
7263:
7243:
7155:
7125:
7082:
7050:
6873:
6839:
6812:
6584:
6453:
6204:
6184:
6164:
6112:
6090:
6016:
5984:
5946:
5904:
5792:
5771:; as a group, it may be identified with the group of
5750:
5730:
5699:
5679:
5653:
5611:
5581:
5552:
5525:
5498:
5473:
5377:
5344:
5302:
5251:
5187:
5133:
5089:
5029:
4892:
4848:
4782:
4747:
4702:
4676:
4636:
4599:
4560:
4525:
4482:
4456:
4416:
4379:
4351:
4329:
4303:
4263:
4223:
4201:
4179:
4126:
4100:
4074:
4034:
4002:
3979:
3959:
3939:
3919:
3892:
3865:
3845:
3796:
3776:
3753:
3727:
3698:
3678:
3642:
3618:
3598:
3578:
3558:
3538:
3518:
3482:
3436:
3194:
3087:
3055:
3049:
is a two-dimensional matrix Lie group, consisting of
3006:
2850:
2820:
2772:
2732:
2682:
2540:
2493:
2453:
2375:
2335:{\displaystyle \mu :G\times G\to G\quad \mu (x,y)=xy}
2282:
2090:
1730:
1708:
1680:
1638:
1598:
1578:
1512:
1157:
1132:
1095:
11447:
Elements of mathematics: Lie groups and Lie algebras
10999:
10954:
9567:{\displaystyle \phi (\exp(x))=\exp(\phi _{*}(x)).\,}
8119:, then the exponential map takes the Lie algebra of
7107:
at the identity. If we identify the Lie algebras of
2137:
is a Lie group of "local" symmetries of a manifold.
2022:) is a Lie group under complex multiplication: the
1500:
11064:
11062:
10942:
10927:
10516:
for the largest connected normal nilpotent subgroup
9779:admits a unique smooth structure which makes it an
8892:, it is common to call elements of the Lie algebra
8744:with multiplication), for complex numbers (because
7716:, on the other hand, is not simply connected. (See
6343:) of square matrices with the Lie bracket given by
5214:{\displaystyle \operatorname {GL} (n,\mathbb {C} )}
5160:{\displaystyle \operatorname {GL} (n,\mathbb {C} )}
5116:{\displaystyle \operatorname {GL} (n,\mathbb {C} )}
5056:{\displaystyle \operatorname {GL} (n,\mathbb {C} )}
4875:{\displaystyle \operatorname {SL} (2,\mathbb {C} )}
4160:is a Lie group; Lie groups of this sort are called
1550:(differentiable) as well, one obtains a Lie group.
11381:
10673:
10507:for the largest connected normal solvable subgroup
10314:capturing this structure is called a Lie algebra (
10254:
10234:
10189:
10165:
10145:
10112:
10042:
10022:
10002:
9971:
9951:
9927:
9887:
9838:
9818:
9795:
9771:
9732:
9712:
9688:
9664:
9644:
9566:
9484:
9454:
9407:
9373:
9302:
9282:
9259:
9055:
9023:
8999:
8968:
8948:
8928:
8908:
8884:
8864:
8841:
8799:
8758:
8732:
8710:
8690:
8670:
8638:
8618:
8587:
8527:
8507:
8487:
8464:
8396:
8376:
8356:
8336:
8296:
8262:
8242:
8222:
8198:
8171:
8151:
8131:
8111:
8069:
8049:
8026:
7919:
7877:
7832:
7754:
7732:, the spin group (double cover of rotation group)
7685:
7665:
7638:
7605:
7573:
7553:
7513:
7489:
7465:
7445:
7418:
7398:
7372:
7269:
7249:
7191:
7138:
7095:
7068:
6883:
6849:
6825:
6620:
6558:
6232:
6190:
6170:
6125:
6098:
6038:
6002:
5964:
5932:
5798:
5763:
5736:
5716:
5685:
5665:
5626:
5593:
5567:
5538:
5511:
5489:(with the group operation being addition) and the
5481:
5388:
5355:
5319:
5288:
5213:
5159:
5115:
5055:
4900:
4874:
4808:
4768:
4733:
4688:
4662:
4622:
4578:
4546:
4511:
4468:
4442:
4402:
4359:
4337:
4315:
4289:
4249:
4209:
4187:
4152:
4108:
4086:
4060:
4008:
3985:
3965:
3945:
3925:
3905:
3878:
3851:
3828:
3782:
3759:
3739:
3713:
3684:
3657:
3624:
3604:
3584:
3564:
3544:
3524:
3497:
3458:
3419:
3163:
3067:
3032:
2986:
2826:
2798:
2758:
2697:
2658:
2520:
2479:
2412:
2334:
2148:) are especially important. Representation theory
2105:
1738:
1716:
1692:
1664:
1624:
1584:
1537:, whereas groups define the abstract concept of a
1165:
1140:
1103:
11842:Lie Groups: An Introduction Through Linear Groups
11803:
11122:. Tata Institute of Fundamental Research, Bombay.
10646:of a Lie group acts transitively on the Lie group
10173:. (One important example of such a system is the
9826:to be a torus of dimension 2 or greater, and let
7146:is a map between the corresponding Lie algebras:
7000:
2423:be a smooth mapping of the product manifold into
1782:ideas were developed in close collaboration with
12934:
11958:. Undergraduate Texts in Mathematics. Springer.
11059:
11023:
6911:We could also define a Lie algebra structure on
5412:; so the group operations are real-analytic and
4922:, a topological group which is also an analytic
4748:
4526:
2618:
539:Representation theory of semisimple Lie algebras
11858:
11762:
11102:
10777:. If the manifold is a circle these are called
10520:so that we have a sequence of normal subgroups
6574:. It can then be shown that the Lie algebra of
6335:) of invertible matrices is the vector space M(
5451:are (roughly) groups of matrices (for example,
1899:Geometric theory and the explicit solutions of
11160:, Proposition 3.30 and Exercise 8 in Chapter 3
10266:and the classification leads to a substantial
8364:is a one-parameter subgroup means simply that
7345:
7013:are Lie groups, then a Lie group homomorphism
5428:
3836:is defined as the length of the shortest path
1794:, published in 1888, 1890, and 1893. The term
12071:
11818:Bulletin of the American Mathematical Society
11787:"Theorie der Transformations-Gruppen (I, II)"
10751:; very little about their structure is known.
8983:of every connected Lie group, because of the
8842:{\displaystyle \mathrm {GL} (n,\mathbb {R} )}
8112:{\displaystyle \mathrm {GL} (n;\mathbb {C} )}
7920:{\displaystyle \mathrm {GL} (n;\mathbb {C} )}
7878:{\displaystyle \mathrm {GL} (n;\mathbb {C} )}
6667:is any group acting smoothly on the manifold
6372:can be thought of informally as the matrices
6059:The product of two Lie groups is a Lie group.
4323:matrices with determinant one and entries in
3181:We now present an example of a group with an
1665:{\displaystyle {\text{GL}}_{n}(\mathbb {C} )}
1625:{\displaystyle {\text{GL}}_{n}(\mathbb {R} )}
1467:
685:
11541:
10875:This is the statement that a Lie group is a
8985:Baker–Campbell–Hausdorff formula
7833:{\displaystyle \mathrm {M} (n;\mathbb {C} )}
6678:is the underlying space of a Lie group
6550:
6472:
5933:{\displaystyle {\text{Sp}}(2n,\mathbb {R} )}
2041:, in contrast with the case of more general
11859:Sattinger, David H.; Weaver, O. L. (1986).
10498:for the connected component of the identity
8297:{\displaystyle c:\mathbb {R} \rightarrow G}
8183:that works for all Lie groups, as follows.
7712:, which as a manifold is the 3-sphere. The
6285:The Lie algebra associated with a Lie group
6240:. These are not Lie groups as they are not
6106:is the universal cover of the circle group
6010:. It is a connected Lie group of dimension
5331:satisfies the above topological definition.
5229:following) but is done roughly as follows:
3747:are disconnected in the subset topology on
2248:
2060:Lie groups play an enormous role in modern
1963:of compact and semisimple Lie groups using
12078:
12064:
10273:The case of a connected compact Lie group
10010:and the subalgebras of the Lie algebra of
9888:{\displaystyle \varphi :\mathbb {R} \to G}
7307:is any discrete subgroup of the center of
4956:which turns it into a Lie group (see also
2443:form a group under multiplication, called
2273:. Smoothness of the group multiplication
2171:On a "global" level, whenever a Lie group
2117:corresponds to enlarging the group to the
1986:enunciating the distinction between Lie's
1474:
1460:
692:
678:
577:Particle physics and representation theory
119:
11950:
11830:
11726:. New York: Academic Press. p. 131.
11692:
11481:, Princeton: Princeton University Press,
11476:
11091:
10990:
10613:is isomorphic to a product of copies of
9928:{\displaystyle \mathrm {im} (\varphi )=H}
9875:
9563:
9218:
9175:
9088:
8832:
8790:
8752:
8726:
8284:
8102:
7910:
7868:
7823:
7406:can be shrunk continuously to a point in
7281:Lie group versus Lie algebra isomorphisms
6545:
6494:
6092:
5987:
5923:
5614:
5475:
5462:
5400:and then, through the local isomorphism,
5279:
5204:
5150:
5106:
5046:
5002:
4968:, and these give most of the examples of
4894:
4865:
4799:
4653:
4613:
4433:
4393:
4353:
4331:
4280:
4240:
4203:
4181:
4143:
4102:
4051:
3701:
3645:
3485:
3452:
3444:
3405:
3391:
3302:
3288:
3280:
3154:
3146:
3023:
2972:
2956:
2948:
2867:
2789:
2749:
2685:
2617:
2557:
2511:
2470:
2129:. This idea later led to the notion of a
2093:
1732:
1710:
1655:
1615:
1159:
1134:
1097:
93:Learn how and when to remove this message
12085:
11839:
11721:
11515:
11321:
11132:
11080:
11041:
10792:topological properties: see for example
10120:. Assume the system in question has the
9374:{\displaystyle \exp(X)\exp(Y)=\exp(X+Y)}
6420:) = 1, which is equivalent to
3667:
2676:real Lie group; it is an open subset of
2001:
1980:International Congress of Mathematicians
1568:Lie groups were first found by studying
56:This article includes a list of general
11792:Archiv for Mathematik og Naturvidenskab
11676:
11595:
11017:
11005:
10960:
10948:
10936:
8740:is the Lie algebra of the Lie group of
6660:of any two derivations is a derivation.
6428: = 0 because Δ = 0.
5817:is a 6-dimensional Lie group of linear
4734:{\displaystyle R^{\mathrm {T} }=R^{-1}}
3505:that is not a Lie group when given the
2261:that is also a finite-dimensional real
2150:is used extensively in particle physics
2080:corresponds to the choice of the group
1809:of first order and on the equations of
1557:, a celebrated example of which is the
544:Representations of classical Lie groups
14:
12935:
12018:
11980:
11653:
11334:BĂ€uerle, de Kerf & ten Kroode 1997
11114:
10972:
10745:two-dimensional conformal field theory
10235:{\displaystyle {\hat {H}}\psi =E\psi }
10113:{\displaystyle {\hat {H}}\psi =E\psi }
9577:In other words, the following diagram
7776:
7288:The first result in this direction is
5637:
5245:) constructs an immersed Lie subgroup
1032:Classification of finite simple groups
12059:
11915:
11893:
11740:
11680:Emergence of the theory of Lie groups
11551:Representation theory. A first course
11406:
11349:
11029:
10879:. For the latter concept, see Bruhat.
10820:Adjoint representation of a Lie group
10303:representations of the Poincaré group
6637:Vector fields on any smooth manifold
3721:. Small neighborhoods of the element
11768:Lie Groups and Representation Theory
11626:
11495:
11441:
11309:
11297:
11253:
11241:
11229:
11217:
11205:
11193:
11181:
11169:
11157:
11144:
11068:
11053:
10862:
10299:representations of the group SL(2,R)
9485:{\displaystyle x\in {\mathfrak {g}}}
7606:{\displaystyle \phi :G\rightarrow H}
7352:Lie groupâLie algebra correspondence
7076:be a Lie group homomorphism and let
6857:is given explicitly by =
6308:The Lie algebra of the vector space
6291:Lie groupâLie algebra correspondence
5239:Lie groupâLie algebra correspondence
4094:invertible matrices with entries in
4023:
2521:{\displaystyle GL_{2}(\mathbb {R} )}
2413:{\displaystyle (x,y)\mapsto x^{-1}y}
1837:that had developed in the theory of
1766:to model the discrete symmetries of
397:Lie groupâLie algebra correspondence
42:
11784:
10800:, for example, a 10-dimensional SU(
10665:and the product of itself with the
9477:
9447:
9437:
9016:
8663:
8611:
8215:
7931:, given by the usual power series:
7762:, and the compact symplectic group
7546:
7536:
7506:
7482:
7356:Fundamental group § Lie groups
7323:have the same Lie algebra (see the
7181:
7171:
6876:
6842:
6815:
5416:itself is a real-analytic manifold.
5181:(For example, a closed subgroup of
4827:
2265:, in which the group operations of
24:
10741:Virasoro algebra from Witt algebra
10053:
9906:
9903:
9583:
8818:
8815:
8088:
8085:
7896:
7893:
7854:
7851:
7809:
7561:is a Lie algebra homomorphism. If
7069:{\displaystyle \phi \colon G\to H}
6210:
6141:
5836:isometries of the Minkowski space.
5673:unitary matrices with determinant
4809:{\displaystyle SO(n,\mathbb {R} )}
4709:
4663:{\displaystyle SO(n,\mathbb {R} )}
4443:{\displaystyle SU(n,\mathbb {C} )}
4290:{\displaystyle SL(n,\mathbb {C} )}
4250:{\displaystyle SL(n,\mathbb {R} )}
4153:{\displaystyle GL(n,\mathbb {C} )}
4061:{\displaystyle GL(n,\mathbb {C} )}
3033:{\displaystyle SO(2,\mathbb {R} )}
2834:as a parameter, this group can be
2799:{\displaystyle SO(2,\mathbb {R} )}
2759:{\displaystyle GL(2,\mathbb {R} )}
2480:{\displaystyle GL(2,\mathbb {R} )}
2014:1 (corresponding to points on the
1974:challenged Lie theorists with his
1792:Theorie der Transformationsgruppen
1533:is a space that locally resembles
62:it lacks sufficient corresponding
25:
12964:
12032:
11746:Lie Groups Beyond an Introduction
10308:
8800:{\displaystyle M(n,\mathbb {R} )}
8465:{\displaystyle c(s+t)=c(s)c(t)\ }
7789:derivative of the exponential map
7473:are Lie groups with Lie algebras
6279:
6146:Some examples of groups that are
6003:{\displaystyle \mathbb {R} ^{2n}}
5832:is a 10-dimensional Lie group of
5237:in the usual manifold sense, the
5007:A Lie group can be defined as a (
4838:is defined in the same way using
4623:{\displaystyle O(n,\mathbb {R} )}
4403:{\displaystyle U(n,\mathbb {C} )}
4017:
3448:
2430:
2175:on a geometric object, such as a
2037:and as such can be studied using
12038:
11521:A History of Geometrical Methods
9854:, i.e. one that winds around in
9415:is a Lie group homomorphism and
7771:fundamental groups of Lie groups
6826:{\displaystyle {\mathfrak {g}}.}
6233:{\displaystyle C^{\infty }(X,G)}
6050:
5627:{\displaystyle \mathbb {R} ^{2}}
4842:rather than real ones (example:
3829:{\displaystyle h_{1},h_{2}\in H}
3714:{\displaystyle \mathbb {T} ^{2}}
3658:{\displaystyle \mathbb {T} ^{2}}
3498:{\displaystyle \mathbb {T} ^{2}}
2698:{\displaystyle \mathbb {R} ^{4}}
2445:general linear group of degree 2
2230:, an important branch of modern
2106:{\displaystyle \mathbb {R} ^{3}}
2018:of center 0 and radius 1 in the
1496:
724:
47:
11920:, World Scientific Publishing,
11832:10.1090/s0002-9904-1959-10358-x
11327:
11303:
11291:
11259:
11247:
11235:
11223:
11211:
11199:
11187:
11175:
11163:
11150:
11138:
11135:, Ch. II, § 2, Proposition 2.7.
11126:
11108:
11096:
11085:
11074:
11047:
10882:
10869:
10674:Infinite-dimensional Lie groups
9624:
9024:{\displaystyle {\mathfrak {g}}}
8671:{\displaystyle {\mathfrak {g}}}
8619:{\displaystyle {\mathfrak {g}}}
8588:{\displaystyle \exp(X)=c(1).\ }
8223:{\displaystyle {\mathfrak {g}}}
7514:{\displaystyle {\mathfrak {h}}}
7490:{\displaystyle {\mathfrak {g}}}
7119:at the identity elements, then
6884:{\displaystyle {\mathfrak {g}}}
6850:{\displaystyle {\mathfrak {g}}}
4593:and special orthogonal groups,
3133:
2304:
2162:the special unitary group SU(3)
2064:, on several different levels.
1881:required to express solutions.
1867:ordinary differential equations
12118:Differentiable/Smooth manifold
11035:
10966:
10902:
10214:
10137:
10092:
9916:
9910:
9879:
9557:
9554:
9548:
9535:
9523:
9520:
9514:
9505:
9442:
9399:
9368:
9356:
9344:
9338:
9329:
9323:
9240:
9231:
9219:
9215:
9197:
9188:
9176:
9172:
9154:
9142:
9101:
9095:
9085:
9079:
8987:: there exists a neighborhood
8956:is the identity component of
8836:
8822:
8794:
8780:
8602:, and it maps the Lie algebra
8576:
8570:
8561:
8555:
8456:
8450:
8444:
8438:
8429:
8417:
8325:
8319:
8288:
8106:
8092:
7953:
7947:
7914:
7900:
7872:
7858:
7827:
7813:
7597:
7541:
7176:
7060:
7001:Homomorphisms and isomorphisms
6597:
6585:
6521:
6512:
6502:
6498:
6484:
6466:
6460:
6316:with the Lie bracket given by
6227:
6215:
5927:
5910:
5717:{\displaystyle {\text{SU}}(2)}
5711:
5705:
5562:
5556:
5363:that is locally isomorphic to
5283:
5269:
5208:
5194:
5154:
5140:
5110:
5096:
5050:
5036:
4869:
4855:
4803:
4789:
4757:
4751:
4657:
4643:
4617:
4603:
4573:
4567:
4535:
4529:
4437:
4423:
4397:
4383:
4284:
4270:
4244:
4230:
4147:
4133:
4055:
4041:
3572:, for example, the portion of
3176:
3027:
3013:
2871:
2857:
2793:
2779:
2753:
2739:
2561:
2547:
2515:
2507:
2474:
2460:
2391:
2388:
2376:
2320:
2308:
2298:
1833:, to complement the theory of
1807:partial differential equations
1659:
1651:
1619:
1611:
1393:Infinite dimensional Lie group
592:Galilean group representations
587:Poincaré group representations
13:
1:
12026:. University of Pennsylvania.
11555:Graduate Texts in Mathematics
11418:American Mathematical Society
11342:
10845:Symmetry in quantum mechanics
10840:Representations of Lie groups
10268:simplification of the problem
10060:Representation of a Lie group
8849:of all invertible matrices).
6981:, so its derivative yields a
6932:The Lie algebra structure on
4966:Lie groups over finite fields
3047:affine group of one dimension
1914:that emerged in the works of
582:Lorentz group representations
549:Theorem of the highest weight
10895:
10472:of any Lie group is an open
9858:. Then there is a Lie group
9408:{\displaystyle \phi :G\to H}
8759:{\displaystyle \mathbb {C} }
8733:{\displaystyle \mathbb {R} }
8250:(i.e., the tangent space to
7783:Exponential map (Lie theory)
7705:simply connected Lie group.
6908:corresponding Lie algebras.
6750:denotes the differential of
6099:{\displaystyle \mathbb {R} }
5482:{\displaystyle \mathbb {R} }
5367:. Then, by a version of the
5083:is the subspace topology of
5023:of the general linear group
4901:{\displaystyle \mathbb {Q} }
4512:{\displaystyle U^{*}=U^{-1}}
4476:complex matrices satisfying
4373:and special unitary groups,
4360:{\displaystyle \mathbb {C} }
4338:{\displaystyle \mathbb {R} }
4210:{\displaystyle \mathbb {C} }
4188:{\displaystyle \mathbb {R} }
4109:{\displaystyle \mathbb {C} }
2144:of the Lie group (or of its
2049:object, the group, with its
1903:of mechanics, worked out by
1739:{\displaystyle \mathbb {C} }
1717:{\displaystyle \mathbb {R} }
1166:{\displaystyle \mathbb {Z} }
1141:{\displaystyle \mathbb {Z} }
1104:{\displaystyle \mathbb {Z} }
7:
12824:Classification of manifolds
11722:Helgason, Sigurdur (1978).
11103:Kobayashi & Oshima 2005
10812:
10591:simple connected Lie groups
7639:{\displaystyle \phi _{*}=f}
7346:Simply connected Lie groups
7218:, we then have a covariant
6384:) such that 1 + Δ
6360:is a closed subgroup of GL(
6249:totally disconnected groups
5965:{\displaystyle 2n\times 2n}
5429:More examples of Lie groups
5423:the topology of a Lie group
3612:is disconnected. The group
2814:. Using the rotation angle
2672:This is a four-dimensional
2349:is a smooth mapping of the
1997:
1947:, led to classification of
1746:. These are now called the
891:List of group theory topics
10:
12969:
11916:Steeb, Willi-Hans (2007),
11766:; Oshima, Toshio. (2005),
11602:Cambridge University Press
11477:Chevalley, Claude (1946),
10749:regular Fréchet Lie groups
10412:A first key result is the
10146:{\displaystyle {\hat {H}}}
10070:Lie algebra representation
10063:
10057:
7786:
7780:
7349:
7225:Two Lie groups are called
6368:) then the Lie algebra of
6288:
6271:for some positive integer
5546:group is often denoted as
5432:
5017:immersely linear Lie group
2214:defined over an arbitrary
2198:In the 1940s–1950s,
1871:differential Galois theory
1773:
534:Lie algebra representation
31:
12900:over commutative algebras
12857:
12816:
12749:
12646:
12542:
12489:
12480:
12316:
12239:
12178:
12098:
12019:Ziller, Wolfgang (2010).
11995:10.1007/978-1-4757-1799-0
11981:Warner, Frank W. (1983),
11964:10.1007/978-0-387-78214-0
11869:10.1007/978-1-4757-1910-9
11694:10.1007/978-1-4612-1202-7
11654:Harvey, F. Reese (1990).
11637:10.1007/978-3-319-13467-3
11563:10.1007/978-1-4612-0979-9
10370:is the largest of these.
9988:one-to-one correspondence
9672:is a Lie group that is a
8539:justifies the definition
7666:{\displaystyle \phi _{*}}
7139:{\displaystyle \phi _{*}}
7096:{\displaystyle \phi _{*}}
6318: = 0.
5666:{\displaystyle 2\times 2}
5594:{\displaystyle 1\times 1}
5439:List of simple Lie groups
4769:{\displaystyle \det(R)=1}
4696:real matrices satisfying
4689:{\displaystyle n\times n}
4547:{\displaystyle \det(U)=1}
4469:{\displaystyle n\times n}
4316:{\displaystyle n\times n}
4087:{\displaystyle n\times n}
3068:{\displaystyle 2\times 2}
1910:The new understanding of
1693:{\displaystyle n\times n}
12616:Riemann curvature tensor
11770:(in Japanese), Iwanami,
11677:Hawkins, Thomas (2000),
11656:Spinors and calibrations
11610:10.1017/CBO9780511791390
11596:Gilmore, Robert (2008).
10857:
10835:List of Lie group topics
10783:Kac–Moody algebras
10767:pointwise multiplication
10609:is abelian. A connected
10458:(for either definition).
9056:{\displaystyle X,Y\in U}
8894:infinitesimal generators
8077:is a closed subgroup of
7204:Lie algebra homomorphism
7202:which turns out to be a
6833:Thus the Lie bracket on
6039:{\displaystyle 2n^{2}+n}
5806:, playing a key role in
5071:of the identity element
4958:HilbertâSmith conjecture
2827:{\displaystyle \varphi }
2249:Definitions and examples
2154:the rotation group SO(3)
2035:differentiable manifolds
1009:Elementary abelian group
886:Glossary of group theory
529:Lie group representation
32:Not to be confused with
11840:Rossmann, Wulf (2001),
11627:Hall, Brian C. (2015),
11517:Coolidge, Julian Lowell
10973:Tresse, Arthur (1893).
9959:will be a sub-torus in
9007:of the zero element of
8337:{\displaystyle c'(0)=X}
7755:{\displaystyle n\geq 3}
7673:is the differential of
6803:, usually denoted by a
6570:) is defined using the
6400:) consists of matrices
6323:The Lie algebra of the
6068:Closed subgroup theorem
5786:Lie group of dimension
5408:can be given by formal
5369:closed subgroup theorem
4934:Hilbert's fifth problem
3946:{\displaystyle \theta }
3672:A portion of the group
3509:. If we take any small
3473:, is a subgroup of the
1949:semisimple Lie algebras
1524:differentiable manifold
554:BorelâWeilâBott theorem
77:more precise citations.
12408:Manifold with boundary
12123:Differential structure
11355:Lectures on Lie Groups
10910:"What is a Lie group?"
10256:
10236:
10191:
10167:
10147:
10114:
10079:in quantum mechanics,
10044:
10024:
10004:
9973:
9953:
9929:
9889:
9848:one-parameter subgroup
9840:
9820:
9797:
9773:
9734:
9714:
9690:
9666:
9646:
9593:natural transformation
9588:
9568:
9486:
9456:
9409:
9375:
9304:
9284:
9261:
9057:
9025:
9001:
8970:
8950:
8930:
8910:
8886:
8866:
8843:
8801:
8760:
8734:
8712:
8692:
8678:and a neighborhood of
8672:
8640:
8620:
8589:
8529:
8509:
8489:
8466:
8398:
8378:
8358:
8338:
8298:
8264:
8244:
8224:
8200:
8173:
8153:
8133:
8113:
8071:
8051:
8028:
7921:
7879:
7834:
7756:
7687:
7667:
7640:
7607:
7575:
7555:
7515:
7491:
7467:
7447:
7420:
7400:
7374:
7271:
7251:
7214:). In the language of
7206:(meaning that it is a
7193:
7140:
7097:
7070:
6885:
6851:
6827:
6788:. This identifies the
6622:
6621:{\displaystyle =XY-YX}
6560:
6345: =
6234:
6192:
6172:
6127:
6100:
6040:
6004:
5972:matrices preserving a
5966:
5934:
5841:exceptional Lie groups
5800:
5765:
5738:
5718:
5687:
5667:
5628:
5595:
5569:
5540:
5513:
5483:
5463:Dimensions one and two
5398:real-analytic manifold
5390:
5357:
5321:
5290:
5215:
5161:
5117:
5067:for some neighborhood
5057:
5003:Topological definition
4902:
4876:
4810:
4770:
4735:
4690:
4664:
4624:
4580:
4548:
4513:
4470:
4444:
4404:
4361:
4339:
4317:
4291:
4251:
4211:
4189:
4154:
4110:
4088:
4062:
4010:
3987:
3973:. With this topology,
3967:
3947:
3927:
3907:
3880:
3853:
3830:
3784:
3767:
3761:
3741:
3740:{\displaystyle h\in H}
3715:
3686:
3659:
3626:
3606:
3586:
3566:
3546:
3526:
3499:
3460:
3421:
3165:
3069:
3034:
2988:
2828:
2800:
2760:
2699:
2660:
2522:
2481:
2414:
2336:
2107:
2027:
1901:differential equations
1851:differential equations
1827:orthogonal polynomials
1760:differential equations
1740:
1718:
1694:
1666:
1626:
1586:
1425:Linear algebraic group
1167:
1142:
1105:
452:Semisimple Lie algebra
407:Adjoint representation
18:Lie group homomorphism
12051:Journal of Lie Theory
11466:, Chapters 7–9
11458:, Chapters 4–6
11450:. Chapters 1–3
10257:
10237:
10192:
10190:{\displaystyle \psi }
10168:
10166:{\displaystyle \psi }
10148:
10115:
10045:
10025:
10005:
9974:
9954:
9930:
9890:
9841:
9821:
9798:
9774:
9735:
9715:
9691:
9667:
9647:
9587:
9569:
9487:
9457:
9410:
9376:
9305:
9285:
9262:
9058:
9026:
9002:
8981:local group structure
8971:
8951:
8931:
8911:
8887:
8867:
8844:
8802:
8761:
8742:positive real numbers
8735:
8713:
8693:
8673:
8641:
8621:
8590:
8530:
8510:
8490:
8467:
8399:
8384:is a smooth map into
8379:
8359:
8339:
8299:
8265:
8245:
8225:
8201:
8174:
8154:
8134:
8114:
8072:
8052:
8029:
7922:
7880:
7835:
7801:from the Lie algebra
7757:
7688:
7686:{\displaystyle \phi }
7668:
7641:
7608:
7576:
7556:
7516:
7492:
7468:
7448:
7421:
7401:
7375:
7272:
7252:
7194:
7141:
7098:
7071:
6895:. The Lie algebra of
6886:
6852:
6828:
6694:by left translations
6641:can be thought of as
6623:
6561:
6443:), may be computed as
6412:with (1 + Δ
6235:
6193:
6173:
6128:
6126:{\displaystyle S^{1}}
6101:
6041:
6005:
5967:
5935:
5801:
5766:
5764:{\displaystyle S^{3}}
5739:
5719:
5688:
5668:
5629:
5596:
5570:
5541:
5539:{\displaystyle S^{1}}
5514:
5512:{\displaystyle S^{1}}
5484:
5391:
5358:
5322:
5291:
5216:
5177:connected components.
5162:
5118:
5058:
4903:
4877:
4811:
4771:
4736:
4691:
4665:
4625:
4581:
4579:{\displaystyle SU(n)}
4549:
4514:
4471:
4445:
4405:
4362:
4340:
4318:
4292:
4252:
4212:
4190:
4171:special linear groups
4155:
4111:
4089:
4063:
4011:
3988:
3968:
3953:in the definition of
3948:
3928:
3908:
3906:{\displaystyle h_{2}}
3881:
3879:{\displaystyle h_{1}}
3854:
3831:
3785:
3762:
3742:
3716:
3687:
3671:
3660:
3627:
3607:
3587:
3567:
3547:
3527:
3500:
3461:
3422:
3166:
3070:
3035:
2989:
2829:
2801:
2761:
2700:
2661:
2523:
2482:
2415:
2337:
2193:representation theory
2108:
2039:differential calculus
2005:
1951:, Cartan's theory of
1756:transformation groups
1741:
1719:
1695:
1667:
1627:
1587:
1168:
1143:
1106:
521:Representation theory
12555:Covariant derivative
12106:Topological manifold
12047:at Wikimedia Commons
11785:Lie, Sophus (1876),
11764:Kobayashi, Toshiyuki
11527:. pp. 304â317.
11479:Theory of Lie groups
11279:on 28 September 2011
10771:quantum field theory
10644:diffeomorphism group
10246:
10205:
10181:
10157:
10128:
10122:rotation group SO(3)
10083:
10077:Schrödinger equation
10034:
10014:
9994:
9963:
9943:
9899:
9865:
9830:
9810:
9787:
9763:
9724:
9704:
9680:
9656:
9636:
9591:(In short, exp is a
9499:
9466:
9419:
9387:
9314:
9294:
9274:
9070:
9035:
9011:
8991:
8960:
8940:
8920:
8900:
8876:
8856:
8811:
8774:
8748:
8722:
8702:
8682:
8658:
8630:
8606:
8546:
8537:exponential function
8519:
8499:
8479:
8411:
8388:
8368:
8348:
8308:
8274:
8254:
8234:
8210:
8190:
8163:
8143:
8123:
8081:
8061:
8041:
7938:
7889:
7847:
7842:general linear group
7805:
7740:
7714:rotation group SO(3)
7677:
7650:
7617:
7585:
7565:
7525:
7501:
7477:
7457:
7437:
7410:
7390:
7364:
7261:
7241:
7210:which preserves the
7153:
7123:
7080:
7048:
6871:
6837:
6810:
6582:
6451:
6325:general linear group
6202:
6182:
6162:
6110:
6088:
6064:topologically closed
6014:
5982:
5944:
5902:
5790:
5748:
5728:
5697:
5677:
5651:
5609:
5579:
5568:{\displaystyle U(1)}
5550:
5523:
5496:
5471:
5375:
5342:
5320:{\displaystyle G,G'}
5300:
5249:
5185:
5131:
5087:
5027:
4970:finite simple groups
4930:-adic neighborhood.
4890:
4846:
4780:
4745:
4700:
4674:
4634:
4597:
4558:
4523:
4480:
4454:
4414:
4377:
4349:
4327:
4301:
4261:
4221:
4199:
4177:
4124:
4098:
4072:
4068:denote the group of
4032:
4016:is an example of a "
4000:
3977:
3957:
3937:
3917:
3913:. In this topology,
3890:
3863:
3843:
3794:
3774:
3751:
3725:
3696:
3676:
3640:
3616:
3596:
3576:
3556:
3536:
3516:
3480:
3434:
3192:
3085:
3053:
3004:
2848:
2818:
2770:
2730:
2680:
2538:
2491:
2451:
2373:
2280:
2220:finite simple groups
2088:
1988:infinitesimal groups
1879:indefinite integrals
1865:, the whole area of
1859:polynomial equations
1728:
1706:
1678:
1636:
1596:
1576:
1155:
1130:
1093:
12589:Exterior derivative
12191:AtiyahâSinger index
12140:Riemannian manifold
11863:. Springer-Verlag.
11115:Bruhat, F. (1958).
10769:), and is used in
10765:(with operation of
10655:orientable manifold
10649:Every Lie group is
10340:simple Lie algebras
10333:and some number of
10331:abelian Lie algebra
8626:into the Lie group
8598:This is called the
8206:in the Lie algebra
7777:The exponential map
7698:Lie's third theorem
7325:table of Lie groups
7290:Lie's third theorem
6652: −
6532: for all
6349: −
5638:Additional examples
5435:Table of Lie groups
5243:Lie's third theorem
4908:, one can define a
2441:invertible matrices
2181:symplectic manifold
2123:projective geometry
1811:classical mechanics
1805:, on the theory of
1768:algebraic equations
1700:invertible matrices
1555:continuous symmetry
799:Group homomorphisms
709:Algebraic structure
666:Table of Lie groups
507:Compact Lie algebra
12895:Secondary calculus
12849:Singularity theory
12804:Parallel transport
12572:De Rham cohomology
12211:Generalized Stokes
11895:Serre, Jean-Pierre
11525:Dover Publications
10992:10.1007/bf02418270
10850:Lie point symmetry
10659:bundle isomorphism
10470:identity component
10422:compact Lie groups
10414:Levi decomposition
10287:The classification
10252:
10232:
10187:
10163:
10143:
10110:
10040:
10020:
10000:
9969:
9949:
9925:
9885:
9836:
9816:
9793:
9769:
9749:group homomorphism
9730:
9710:
9696:and such that the
9686:
9662:
9642:
9589:
9564:
9482:
9452:
9405:
9371:
9300:
9280:
9257:
9213:
9170:
9140:
9053:
9021:
8997:
8966:
8946:
8926:
8916:. The subgroup of
8906:
8882:
8862:
8839:
8797:
8756:
8730:
8708:
8688:
8668:
8636:
8616:
8585:
8525:
8505:
8485:
8462:
8394:
8374:
8354:
8334:
8294:
8260:
8240:
8220:
8196:
8169:
8149:
8129:
8109:
8067:
8047:
8024:
7929:matrix exponential
7927:is defined by the
7917:
7875:
7830:
7793:normal coordinates
7752:
7683:
7663:
7636:
7603:
7571:
7551:
7511:
7487:
7463:
7443:
7416:
7396:
7370:
7267:
7247:
7229:if there exists a
7189:
7136:
7093:
7066:
7027:group homomorphism
6983:bilinear operation
6905:locally isomorphic
6881:
6847:
6823:
6618:
6572:matrix exponential
6556:
6242:finite-dimensional
6230:
6188:
6168:
6123:
6096:
6036:
6000:
5962:
5930:
5796:
5761:
5734:
5714:
5683:
5663:
5624:
5601:unitary matrices.
5591:
5565:
5536:
5509:
5479:
5389:{\displaystyle G'}
5386:
5356:{\displaystyle G'}
5353:
5317:
5286:
5233:Given a Lie group
5211:
5157:
5113:
5079:, the topology on
5053:
4898:
4872:
4806:
4766:
4731:
4686:
4660:
4620:
4576:
4544:
4509:
4466:
4440:
4400:
4357:
4335:
4313:
4287:
4247:
4207:
4185:
4162:matrix Lie groups.
4150:
4106:
4084:
4058:
4006:
3983:
3963:
3943:
3923:
3903:
3876:
3849:
3826:
3780:
3768:
3757:
3737:
3711:
3682:
3655:
3622:
3602:
3582:
3562:
3542:
3522:
3495:
3456:
3417:
3382:
3271:
3161:
3124:
3065:
3030:
2984:
2939:
2824:
2796:
2756:
2695:
2656:
2608:
2518:
2477:
2410:
2332:
2269:and inversion are
2224:algebraic geometry
2158:double cover SU(2)
2115:conformal geometry
2103:
2078:Euclidean geometry
2043:topological groups
2028:
1959:'s description of
1853:. On the model of
1841:, in the hands of
1803:Carl Gustav Jacobi
1736:
1714:
1690:
1662:
1622:
1582:
1275:Special orthogonal
1163:
1138:
1101:
982:Lagrange's theorem
438:Affine Lie algebra
428:Simple Lie algebra
169:Special orthogonal
12930:
12929:
12812:
12811:
12577:Differential form
12231:Whitney embedding
12165:Differential form
12043:Media related to
12004:978-0-387-90894-6
11935:978-981-270-809-0
11908:978-3-540-55008-2
11878:978-3-540-96240-3
11851:978-0-19-859683-7
11805:Nijenhuis, Albert
11755:978-0-8176-4259-4
11742:Knapp, Anthony W.
11733:978-0-12-338460-7
11704:978-0-387-98963-1
11619:978-0-521-88400-6
11572:978-0-387-97495-8
11534:978-0-486-49524-8
11497:Cohn, Paul Moritz
11488:978-0-691-04990-8
11443:Bourbaki, Nicolas
11427:978-0-8218-0288-5
11395:978-0-444-82836-1
11364:978-0-226-00527-0
11351:Adams, John Frank
11105:, Definition 5.3.
10863:Explanatory notes
10830:Homogeneous space
10808:becomes infinite.
10733:central extension
10611:abelian Lie group
10587:central extension
10444:Simple Lie groups
10255:{\displaystyle E}
10217:
10140:
10095:
10043:{\displaystyle G}
10023:{\displaystyle G}
10003:{\displaystyle G}
9972:{\displaystyle G}
9952:{\displaystyle H}
9839:{\displaystyle H}
9819:{\displaystyle G}
9796:{\displaystyle G}
9772:{\displaystyle G}
9733:{\displaystyle G}
9713:{\displaystyle H}
9689:{\displaystyle G}
9665:{\displaystyle G}
9645:{\displaystyle H}
9303:{\displaystyle Y}
9283:{\displaystyle X}
9212:
9169:
9139:
9000:{\displaystyle U}
8969:{\displaystyle G}
8949:{\displaystyle N}
8929:{\displaystyle G}
8909:{\displaystyle G}
8885:{\displaystyle e}
8865:{\displaystyle N}
8711:{\displaystyle G}
8691:{\displaystyle e}
8639:{\displaystyle G}
8584:
8528:{\displaystyle G}
8508:{\displaystyle t}
8488:{\displaystyle s}
8461:
8397:{\displaystyle G}
8377:{\displaystyle c}
8357:{\displaystyle c}
8263:{\displaystyle G}
8243:{\displaystyle G}
8199:{\displaystyle X}
8172:{\displaystyle G}
8152:{\displaystyle G}
8132:{\displaystyle G}
8070:{\displaystyle G}
8050:{\displaystyle X}
8016:
7991:
7726:half-integer spin
7718:Topology of SO(3)
7574:{\displaystyle G}
7466:{\displaystyle H}
7446:{\displaystyle G}
7419:{\displaystyle G}
7399:{\displaystyle G}
7386:if every loop in
7373:{\displaystyle G}
7270:{\displaystyle H}
7250:{\displaystyle G}
6867:This Lie algebra
6541:
6533:
6416:)(1 + Δ
6191:{\displaystyle G}
6171:{\displaystyle X}
6152:discrete topology
5908:
5890:simple Lie groups
5808:quantum mechanics
5799:{\displaystyle 3}
5737:{\displaystyle 3}
5703:
5693:. Topologically,
5686:{\displaystyle 1}
5457:symplectic groups
5241:(or a version of
5019:to be a subgroup
5013:topological group
4884:metric completion
4840:complex manifolds
4835:complex Lie group
4591:orthogonal groups
4024:Matrix Lie groups
4009:{\displaystyle H}
3986:{\displaystyle H}
3966:{\displaystyle H}
3926:{\displaystyle H}
3852:{\displaystyle H}
3783:{\displaystyle H}
3760:{\displaystyle H}
3685:{\displaystyle H}
3625:{\displaystyle H}
3605:{\displaystyle U}
3585:{\displaystyle H}
3565:{\displaystyle H}
3545:{\displaystyle h}
3525:{\displaystyle U}
3507:subspace topology
3471:irrational number
2228:automorphic forms
1978:presented at the
1831:continuous groups
1823:special functions
1643:
1603:
1585:{\displaystyle G}
1484:
1483:
1059:
1058:
941:Alternating group
898:
897:
702:
701:
502:Split Lie algebra
465:Cartan subalgebra
327:
326:
218:Simple Lie groups
103:
102:
95:
34:Group of Lie type
16:(Redirected from
12960:
12922:Stratified space
12880:Fréchet manifold
12594:Interior product
12487:
12486:
12184:
12080:
12073:
12066:
12057:
12056:
12042:
12027:
12025:
12015:
11977:
11956:Naive Lie Theory
11946:
11911:
11890:
11854:
11836:
11834:
11813:, by P. M. Cohn"
11800:
11799:: 19â57, 152â193
11780:
11758:
11737:
11715:
11696:
11673:
11649:
11623:
11592:
11538:
11512:
11491:
11449:
11438:
11403:
11387:
11375:
11336:
11331:
11325:
11319:
11313:
11307:
11301:
11295:
11289:
11288:
11286:
11284:
11278:
11271:
11263:
11257:
11251:
11245:
11239:
11233:
11227:
11221:
11215:
11209:
11203:
11197:
11191:
11185:
11179:
11173:
11167:
11161:
11154:
11148:
11142:
11136:
11130:
11124:
11123:
11121:
11112:
11106:
11100:
11094:
11089:
11083:
11078:
11072:
11066:
11057:
11051:
11045:
11039:
11033:
11027:
11021:
11015:
11009:
11003:
10997:
10996:
10994:
10979:Acta Mathematica
10970:
10964:
10958:
10952:
10946:
10940:
10934:
10925:
10924:
10922:
10920:
10906:
10889:
10886:
10880:
10877:formal Lie group
10873:
10794:Kuiper's theorem
10775:Donaldson theory
10737:Virasoro algebra
10690:Banach manifolds
10669:at the identity)
10589:of a product of
10550:
10403:simply connected
10261:
10259:
10258:
10253:
10241:
10239:
10238:
10233:
10219:
10218:
10210:
10201:of solutions to
10196:
10194:
10193:
10188:
10172:
10170:
10169:
10164:
10152:
10150:
10149:
10144:
10142:
10141:
10133:
10119:
10117:
10116:
10111:
10097:
10096:
10088:
10049:
10047:
10046:
10041:
10029:
10027:
10026:
10021:
10009:
10007:
10006:
10001:
9978:
9976:
9975:
9970:
9958:
9956:
9955:
9950:
9934:
9932:
9931:
9926:
9909:
9894:
9892:
9891:
9886:
9878:
9852:irrational slope
9845:
9843:
9842:
9837:
9825:
9823:
9822:
9817:
9802:
9800:
9799:
9794:
9783:Lie subgroup of
9778:
9776:
9775:
9770:
9753:Cartan's theorem
9739:
9737:
9736:
9731:
9719:
9717:
9716:
9711:
9695:
9693:
9692:
9687:
9671:
9669:
9668:
9663:
9651:
9649:
9648:
9643:
9611:
9573:
9571:
9570:
9565:
9547:
9546:
9491:
9489:
9488:
9483:
9481:
9480:
9461:
9459:
9458:
9453:
9451:
9450:
9441:
9440:
9431:
9430:
9414:
9412:
9411:
9406:
9380:
9378:
9377:
9372:
9309:
9307:
9306:
9301:
9289:
9287:
9286:
9281:
9266:
9264:
9263:
9258:
9253:
9249:
9214:
9205:
9171:
9162:
9141:
9132:
9062:
9060:
9059:
9054:
9031:, such that for
9030:
9028:
9027:
9022:
9020:
9019:
9006:
9004:
9003:
8998:
8975:
8973:
8972:
8967:
8955:
8953:
8952:
8947:
8935:
8933:
8932:
8927:
8915:
8913:
8912:
8907:
8891:
8889:
8888:
8883:
8871:
8869:
8868:
8863:
8848:
8846:
8845:
8840:
8835:
8821:
8806:
8804:
8803:
8798:
8793:
8765:
8763:
8762:
8757:
8755:
8739:
8737:
8736:
8731:
8729:
8717:
8715:
8714:
8709:
8697:
8695:
8694:
8689:
8677:
8675:
8674:
8669:
8667:
8666:
8646:. It provides a
8645:
8643:
8642:
8637:
8625:
8623:
8622:
8617:
8615:
8614:
8594:
8592:
8591:
8586:
8582:
8534:
8532:
8531:
8526:
8514:
8512:
8511:
8506:
8494:
8492:
8491:
8486:
8471:
8469:
8468:
8463:
8459:
8403:
8401:
8400:
8395:
8383:
8381:
8380:
8375:
8363:
8361:
8360:
8355:
8343:
8341:
8340:
8335:
8318:
8303:
8301:
8300:
8295:
8287:
8269:
8267:
8266:
8261:
8249:
8247:
8246:
8241:
8229:
8227:
8226:
8221:
8219:
8218:
8205:
8203:
8202:
8197:
8186:For each vector
8178:
8176:
8175:
8170:
8158:
8156:
8155:
8150:
8138:
8136:
8135:
8130:
8118:
8116:
8115:
8110:
8105:
8091:
8076:
8074:
8073:
8068:
8056:
8054:
8053:
8048:
8033:
8031:
8030:
8025:
8017:
8015:
8007:
8006:
7997:
7992:
7990:
7982:
7981:
7972:
7926:
7924:
7923:
7918:
7913:
7899:
7884:
7882:
7881:
7876:
7871:
7857:
7839:
7837:
7836:
7831:
7826:
7812:
7761:
7759:
7758:
7753:
7693:at the identity.
7692:
7690:
7689:
7684:
7672:
7670:
7669:
7664:
7662:
7661:
7645:
7643:
7642:
7637:
7629:
7628:
7612:
7610:
7609:
7604:
7580:
7578:
7577:
7572:
7560:
7558:
7557:
7552:
7550:
7549:
7540:
7539:
7520:
7518:
7517:
7512:
7510:
7509:
7496:
7494:
7493:
7488:
7486:
7485:
7472:
7470:
7469:
7464:
7452:
7450:
7449:
7444:
7425:
7423:
7422:
7417:
7405:
7403:
7402:
7397:
7383:simply connected
7379:
7377:
7376:
7371:
7340:simply connected
7276:
7274:
7273:
7268:
7256:
7254:
7253:
7248:
7198:
7196:
7195:
7190:
7185:
7184:
7175:
7174:
7165:
7164:
7145:
7143:
7142:
7137:
7135:
7134:
7102:
7100:
7099:
7094:
7092:
7091:
7075:
7073:
7072:
7067:
6890:
6888:
6887:
6882:
6880:
6879:
6856:
6854:
6853:
6848:
6846:
6845:
6832:
6830:
6829:
6824:
6819:
6818:
6627:
6625:
6624:
6619:
6565:
6563:
6562:
6557:
6549:
6548:
6542:
6539:
6534:
6531:
6505:
6497:
6265:local properties
6239:
6237:
6236:
6231:
6214:
6213:
6197:
6195:
6194:
6189:
6177:
6175:
6174:
6169:
6132:
6130:
6129:
6124:
6122:
6121:
6105:
6103:
6102:
6097:
6095:
6072:Cartan's theorem
6045:
6043:
6042:
6037:
6029:
6028:
6009:
6007:
6006:
6001:
5999:
5998:
5990:
5971:
5969:
5968:
5963:
5940:consists of all
5939:
5937:
5936:
5931:
5926:
5909:
5906:
5897:symplectic group
5805:
5803:
5802:
5797:
5780:Heisenberg group
5773:unit quaternions
5770:
5768:
5767:
5762:
5760:
5759:
5743:
5741:
5740:
5735:
5723:
5721:
5720:
5715:
5704:
5701:
5692:
5690:
5689:
5684:
5672:
5670:
5669:
5664:
5647:is the group of
5633:
5631:
5630:
5625:
5623:
5622:
5617:
5600:
5598:
5597:
5592:
5574:
5572:
5571:
5566:
5545:
5543:
5542:
5537:
5535:
5534:
5518:
5516:
5515:
5510:
5508:
5507:
5488:
5486:
5485:
5480:
5478:
5449:algebraic groups
5395:
5393:
5392:
5387:
5385:
5362:
5360:
5359:
5354:
5352:
5334:Conversely, let
5326:
5324:
5323:
5318:
5316:
5295:
5293:
5292:
5287:
5282:
5259:
5220:
5218:
5217:
5212:
5207:
5166:
5164:
5163:
5158:
5153:
5122:
5120:
5119:
5114:
5109:
5062:
5060:
5059:
5054:
5049:
4997:groupoid objects
4975:The language of
4962:Hilbert manifold
4907:
4905:
4904:
4899:
4897:
4881:
4879:
4878:
4873:
4868:
4828:Related concepts
4822:classical groups
4815:
4813:
4812:
4807:
4802:
4775:
4773:
4772:
4767:
4740:
4738:
4737:
4732:
4730:
4729:
4714:
4713:
4712:
4695:
4693:
4692:
4687:
4670:, consisting of
4669:
4667:
4666:
4661:
4656:
4629:
4627:
4626:
4621:
4616:
4585:
4583:
4582:
4577:
4553:
4551:
4550:
4545:
4518:
4516:
4515:
4510:
4508:
4507:
4492:
4491:
4475:
4473:
4472:
4467:
4450:, consisting of
4449:
4447:
4446:
4441:
4436:
4409:
4407:
4406:
4401:
4396:
4366:
4364:
4363:
4358:
4356:
4344:
4342:
4341:
4336:
4334:
4322:
4320:
4319:
4314:
4297:, consisting of
4296:
4294:
4293:
4288:
4283:
4256:
4254:
4253:
4248:
4243:
4216:
4214:
4213:
4208:
4206:
4194:
4192:
4191:
4186:
4184:
4159:
4157:
4156:
4151:
4146:
4115:
4113:
4112:
4107:
4105:
4093:
4091:
4090:
4085:
4067:
4065:
4064:
4059:
4054:
4015:
4013:
4012:
4007:
3992:
3990:
3989:
3984:
3972:
3970:
3969:
3964:
3952:
3950:
3949:
3944:
3932:
3930:
3929:
3924:
3912:
3910:
3909:
3904:
3902:
3901:
3885:
3883:
3882:
3877:
3875:
3874:
3858:
3856:
3855:
3850:
3835:
3833:
3832:
3827:
3819:
3818:
3806:
3805:
3789:
3787:
3786:
3781:
3766:
3764:
3763:
3758:
3746:
3744:
3743:
3738:
3720:
3718:
3717:
3712:
3710:
3709:
3704:
3691:
3689:
3688:
3683:
3664:
3662:
3661:
3656:
3654:
3653:
3648:
3631:
3629:
3628:
3623:
3611:
3609:
3608:
3603:
3591:
3589:
3588:
3583:
3571:
3569:
3568:
3563:
3551:
3549:
3548:
3543:
3531:
3529:
3528:
3523:
3504:
3502:
3501:
3496:
3494:
3493:
3488:
3465:
3463:
3462:
3457:
3455:
3447:
3426:
3424:
3423:
3418:
3413:
3409:
3408:
3387:
3383:
3379:
3378:
3346:
3345:
3311:
3310:
3305:
3296:
3292:
3291:
3276:
3272:
3268:
3267:
3232:
3231:
3170:
3168:
3167:
3162:
3157:
3129:
3125:
3074:
3072:
3071:
3066:
3039:
3037:
3036:
3031:
3026:
2993:
2991:
2990:
2985:
2980:
2976:
2975:
2964:
2959:
2944:
2943:
2870:
2833:
2831:
2830:
2825:
2805:
2803:
2802:
2797:
2792:
2765:
2763:
2762:
2757:
2752:
2722:matrices form a
2705:. This group is
2704:
2702:
2701:
2696:
2694:
2693:
2688:
2665:
2663:
2662:
2657:
2652:
2648:
2613:
2612:
2560:
2527:
2525:
2524:
2519:
2514:
2506:
2505:
2486:
2484:
2483:
2478:
2473:
2419:
2417:
2416:
2411:
2406:
2405:
2361:
2351:product manifold
2341:
2339:
2338:
2333:
2226:. The theory of
2222:, as well as in
2212:algebraic groups
2208:Claude Chevalley
2185:Lie group action
2127:projective group
2112:
2110:
2109:
2104:
2102:
2101:
2096:
2070:Erlangen program
1992:Claude Chevalley
1953:symmetric spaces
1886:Bernhard Riemann
1748:classical groups
1745:
1743:
1742:
1737:
1735:
1723:
1721:
1720:
1715:
1713:
1699:
1697:
1696:
1691:
1671:
1669:
1668:
1663:
1658:
1650:
1649:
1644:
1641:
1631:
1629:
1628:
1623:
1618:
1610:
1609:
1604:
1601:
1591:
1589:
1588:
1583:
1544:continuous group
1539:binary operation
1515:
1510:
1509:
1506:
1505:
1502:
1476:
1469:
1462:
1418:Algebraic groups
1191:Hyperbolic group
1181:Arithmetic group
1172:
1170:
1169:
1164:
1162:
1147:
1145:
1144:
1139:
1137:
1110:
1108:
1107:
1102:
1100:
1023:Schur multiplier
977:Cauchy's theorem
965:Quaternion group
913:
912:
739:
738:
728:
715:
704:
703:
694:
687:
680:
637:Claude Chevalley
494:Complexification
337:Other Lie groups
223:
222:
131:Classical groups
123:
105:
104:
98:
91:
87:
84:
78:
73:this article by
64:inline citations
51:
50:
43:
21:
12968:
12967:
12963:
12962:
12961:
12959:
12958:
12957:
12933:
12932:
12931:
12926:
12865:Banach manifold
12858:Generalizations
12853:
12808:
12745:
12642:
12604:Ricci curvature
12560:Cotangent space
12538:
12476:
12318:
12312:
12271:Exponential map
12235:
12180:
12174:
12094:
12084:
12035:
12030:
12023:
12005:
11987:Springer-Verlag
11974:
11952:Stillwell, John
11936:
11909:
11879:
11852:
11778:
11756:
11734:
11705:
11685:Springer-Verlag
11670:
11647:
11620:
11573:
11543:Fulton, William
11535:
11489:
11428:
11396:
11365:
11345:
11340:
11339:
11332:
11328:
11320:
11316:
11308:
11304:
11296:
11292:
11282:
11280:
11276:
11269:
11265:
11264:
11260:
11252:
11248:
11240:
11236:
11228:
11224:
11216:
11212:
11204:
11200:
11192:
11188:
11180:
11176:
11172:Corollary 3.50.
11168:
11164:
11155:
11151:
11143:
11139:
11131:
11127:
11119:
11113:
11109:
11101:
11097:
11090:
11086:
11079:
11075:
11067:
11060:
11052:
11048:
11040:
11036:
11028:
11024:
11016:
11012:
11004:
11000:
10971:
10967:
10959:
10955:
10947:
10943:
10935:
10928:
10918:
10916:
10908:
10907:
10903:
10898:
10893:
10892:
10887:
10883:
10874:
10870:
10865:
10860:
10855:
10815:
10725:diffeomorphisms
10685:Euclidean space
10683:(as opposed to
10676:
10653:, and hence an
10632:
10608:
10601:
10584:
10577:
10568:
10545:
10538:
10531:
10524:
10515:
10506:
10497:
10474:normal subgroup
10430:Dynkin diagrams
10369:
10365:
10359:
10353:
10347:
10322:at each point.
10311:
10247:
10244:
10243:
10209:
10208:
10206:
10203:
10202:
10182:
10179:
10178:
10158:
10155:
10154:
10132:
10131:
10129:
10126:
10125:
10087:
10086:
10084:
10081:
10080:
10072:
10062:
10056:
10054:Representations
10035:
10032:
10031:
10015:
10012:
10011:
9995:
9992:
9991:
9984:exponential map
9964:
9961:
9960:
9944:
9941:
9940:
9902:
9900:
9897:
9896:
9874:
9866:
9863:
9862:
9831:
9828:
9827:
9811:
9808:
9807:
9788:
9785:
9784:
9764:
9761:
9760:
9751:. According to
9725:
9722:
9721:
9705:
9702:
9701:
9681:
9678:
9677:
9657:
9654:
9653:
9652:of a Lie group
9637:
9634:
9633:
9627:
9603:
9542:
9538:
9500:
9497:
9496:
9476:
9475:
9467:
9464:
9463:
9446:
9445:
9436:
9435:
9426:
9422:
9420:
9417:
9416:
9388:
9385:
9384:
9315:
9312:
9311:
9295:
9292:
9291:
9275:
9272:
9271:
9203:
9160:
9130:
9117:
9113:
9071:
9068:
9067:
9036:
9033:
9032:
9015:
9014:
9012:
9009:
9008:
8992:
8989:
8988:
8961:
8958:
8957:
8941:
8938:
8937:
8921:
8918:
8917:
8901:
8898:
8897:
8877:
8874:
8873:
8857:
8854:
8853:
8831:
8814:
8812:
8809:
8808:
8789:
8775:
8772:
8771:
8751:
8749:
8746:
8745:
8725:
8723:
8720:
8719:
8703:
8700:
8699:
8683:
8680:
8679:
8662:
8661:
8659:
8656:
8655:
8631:
8628:
8627:
8610:
8609:
8607:
8604:
8603:
8600:exponential map
8547:
8544:
8543:
8520:
8517:
8516:
8500:
8497:
8496:
8480:
8477:
8476:
8412:
8409:
8408:
8389:
8386:
8385:
8369:
8366:
8365:
8349:
8346:
8345:
8311:
8309:
8306:
8305:
8283:
8275:
8272:
8271:
8255:
8252:
8251:
8235:
8232:
8231:
8214:
8213:
8211:
8208:
8207:
8191:
8188:
8187:
8164:
8161:
8160:
8144:
8141:
8140:
8124:
8121:
8120:
8101:
8084:
8082:
8079:
8078:
8062:
8059:
8058:
8042:
8039:
8038:
8008:
8002:
7998:
7996:
7983:
7977:
7973:
7971:
7939:
7936:
7935:
7909:
7892:
7890:
7887:
7886:
7867:
7850:
7848:
7845:
7844:
7822:
7808:
7806:
7803:
7802:
7799:exponential map
7795:
7785:
7779:
7741:
7738:
7737:
7678:
7675:
7674:
7657:
7653:
7651:
7648:
7647:
7624:
7620:
7618:
7615:
7614:
7586:
7583:
7582:
7566:
7563:
7562:
7545:
7544:
7535:
7534:
7526:
7523:
7522:
7505:
7504:
7502:
7499:
7498:
7481:
7480:
7478:
7475:
7474:
7458:
7455:
7454:
7438:
7435:
7434:
7411:
7408:
7407:
7391:
7388:
7387:
7365:
7362:
7361:
7358:
7348:
7283:
7262:
7259:
7258:
7257:to a Lie group
7242:
7239:
7238:
7216:category theory
7180:
7179:
7170:
7169:
7160:
7156:
7154:
7151:
7150:
7130:
7126:
7124:
7121:
7120:
7087:
7083:
7081:
7078:
7077:
7049:
7046:
7045:
7031:holomorphic map
7003:
6990:
6937:
6927:
6916:
6875:
6874:
6872:
6869:
6868:
6862:
6841:
6840:
6838:
6835:
6834:
6814:
6813:
6811:
6808:
6807:
6796:
6784:
6780:
6774:
6755:
6749:
6745:
6730:
6723:
6718:
6714:
6699:
6583:
6580:
6579:
6544:
6543:
6538:
6530:
6501:
6493:
6452:
6449:
6448:
6344:
6319:
6317:
6298:infinitesimally
6293:
6287:
6282:
6209:
6205:
6203:
6200:
6199:
6183:
6180:
6179:
6178:to a Lie group
6163:
6160:
6159:
6144:
6142:Related notions
6117:
6113:
6111:
6108:
6107:
6091:
6089:
6086:
6085:
6082:universal cover
6053:
6024:
6020:
6015:
6012:
6011:
5991:
5986:
5985:
5983:
5980:
5979:
5975:symplectic form
5945:
5942:
5941:
5922:
5905:
5903:
5900:
5899:
5886:
5877:
5868:
5859:
5850:
5823:Minkowski space
5791:
5788:
5787:
5782:is a connected
5755:
5751:
5749:
5746:
5745:
5729:
5726:
5725:
5700:
5698:
5695:
5694:
5678:
5675:
5674:
5652:
5649:
5648:
5640:
5618:
5613:
5612:
5610:
5607:
5606:
5580:
5577:
5576:
5575:, the group of
5551:
5548:
5547:
5530:
5526:
5524:
5521:
5520:
5503:
5499:
5497:
5494:
5493:
5474:
5472:
5469:
5468:
5465:
5441:
5431:
5378:
5376:
5373:
5372:
5345:
5343:
5340:
5339:
5309:
5301:
5298:
5297:
5278:
5252:
5250:
5247:
5246:
5203:
5186:
5183:
5182:
5149:
5132:
5129:
5128:
5105:
5088:
5085:
5084:
5045:
5028:
5025:
5024:
5005:
4989:Lie supergroups
4977:category theory
4948:showed that if
4913:-adic Lie group
4893:
4891:
4888:
4887:
4864:
4847:
4844:
4843:
4830:
4798:
4781:
4778:
4777:
4776:in the case of
4746:
4743:
4742:
4722:
4718:
4708:
4707:
4703:
4701:
4698:
4697:
4675:
4672:
4671:
4652:
4635:
4632:
4631:
4612:
4598:
4595:
4594:
4559:
4556:
4555:
4554:in the case of
4524:
4521:
4520:
4500:
4496:
4487:
4483:
4481:
4478:
4477:
4455:
4452:
4451:
4432:
4415:
4412:
4411:
4392:
4378:
4375:
4374:
4352:
4350:
4347:
4346:
4330:
4328:
4325:
4324:
4302:
4299:
4298:
4279:
4262:
4259:
4258:
4239:
4222:
4219:
4218:
4202:
4200:
4197:
4196:
4180:
4178:
4175:
4174:
4142:
4125:
4122:
4121:
4118:closed subgroup
4101:
4099:
4096:
4095:
4073:
4070:
4069:
4050:
4033:
4030:
4029:
4026:
4001:
3998:
3997:
3978:
3975:
3974:
3958:
3955:
3954:
3938:
3935:
3934:
3918:
3915:
3914:
3897:
3893:
3891:
3888:
3887:
3870:
3866:
3864:
3861:
3860:
3844:
3841:
3840:
3814:
3810:
3801:
3797:
3795:
3792:
3791:
3775:
3772:
3771:
3752:
3749:
3748:
3726:
3723:
3722:
3705:
3700:
3699:
3697:
3694:
3693:
3677:
3674:
3673:
3649:
3644:
3643:
3641:
3638:
3637:
3617:
3614:
3613:
3597:
3594:
3593:
3577:
3574:
3573:
3557:
3554:
3553:
3537:
3534:
3533:
3517:
3514:
3513:
3489:
3484:
3483:
3481:
3478:
3477:
3451:
3443:
3435:
3432:
3431:
3404:
3381:
3380:
3365:
3361:
3359:
3353:
3352:
3347:
3332:
3328:
3324:
3320:
3319:
3315:
3306:
3301:
3300:
3287:
3270:
3269:
3251:
3247:
3245:
3239:
3238:
3233:
3218:
3214:
3210:
3206:
3205:
3201:
3193:
3190:
3189:
3179:
3153:
3123:
3122:
3117:
3111:
3110:
3105:
3098:
3094:
3086:
3083:
3082:
3054:
3051:
3050:
3022:
3005:
3002:
3001:
2971:
2960:
2955:
2938:
2937:
2926:
2914:
2913:
2899:
2883:
2882:
2881:
2877:
2866:
2849:
2846:
2845:
2819:
2816:
2815:
2788:
2771:
2768:
2767:
2748:
2731:
2728:
2727:
2689:
2684:
2683:
2681:
2678:
2677:
2607:
2606:
2601:
2595:
2594:
2589:
2579:
2578:
2571:
2567:
2556:
2539:
2536:
2535:
2510:
2501:
2497:
2492:
2489:
2488:
2469:
2452:
2449:
2448:
2447:and denoted by
2433:
2398:
2394:
2374:
2371:
2370:
2353:
2281:
2278:
2277:
2263:smooth manifold
2251:
2142:representations
2119:conformal group
2097:
2092:
2091:
2089:
2086:
2085:
2030:Lie groups are
2008:complex numbers
2006:The set of all
2000:
1965:highest weights
1961:representations
1933:Wilhelm Killing
1835:discrete groups
1819:Ăvariste Galois
1788:Friedrich Engel
1776:
1731:
1729:
1726:
1725:
1709:
1707:
1704:
1703:
1679:
1676:
1675:
1654:
1645:
1640:
1639:
1637:
1634:
1633:
1614:
1605:
1600:
1599:
1597:
1594:
1593:
1577:
1574:
1573:
1535:Euclidean space
1522:that is also a
1513:
1499:
1495:
1480:
1451:
1450:
1439:Abelian variety
1432:Reductive group
1420:
1410:
1409:
1408:
1407:
1358:
1350:
1342:
1334:
1326:
1299:Special unitary
1210:
1196:
1195:
1177:
1176:
1158:
1156:
1153:
1152:
1133:
1131:
1128:
1127:
1096:
1094:
1091:
1090:
1082:
1081:
1072:Discrete groups
1061:
1060:
1016:Frobenius group
961:
948:
937:
930:Symmetric group
926:
910:
900:
899:
750:Normal subgroup
736:
716:
707:
698:
653:
652:
651:
622:Wilhelm Killing
606:
598:
597:
596:
571:
560:
559:
558:
523:
513:
512:
511:
498:
482:
460:Dynkin diagrams
454:
444:
443:
442:
424:
402:Exponential map
391:
381:
380:
379:
360:Conformal group
339:
329:
328:
320:
312:
304:
296:
288:
269:
259:
249:
239:
220:
210:
209:
208:
189:Special unitary
133:
99:
88:
82:
79:
69:Please help to
68:
52:
48:
41:
28:
23:
22:
15:
12:
11:
5:
12966:
12956:
12955:
12950:
12945:
12928:
12927:
12925:
12924:
12919:
12914:
12909:
12904:
12903:
12902:
12892:
12887:
12882:
12877:
12872:
12867:
12861:
12859:
12855:
12854:
12852:
12851:
12846:
12841:
12836:
12831:
12826:
12820:
12818:
12814:
12813:
12810:
12809:
12807:
12806:
12801:
12796:
12791:
12786:
12781:
12776:
12771:
12766:
12761:
12755:
12753:
12747:
12746:
12744:
12743:
12738:
12733:
12728:
12723:
12718:
12713:
12703:
12698:
12693:
12683:
12678:
12673:
12668:
12663:
12658:
12652:
12650:
12644:
12643:
12641:
12640:
12635:
12630:
12629:
12628:
12618:
12613:
12612:
12611:
12601:
12596:
12591:
12586:
12585:
12584:
12574:
12569:
12568:
12567:
12557:
12552:
12546:
12544:
12540:
12539:
12537:
12536:
12531:
12526:
12521:
12520:
12519:
12509:
12504:
12499:
12493:
12491:
12484:
12478:
12477:
12475:
12474:
12469:
12459:
12454:
12440:
12435:
12430:
12425:
12420:
12418:Parallelizable
12415:
12410:
12405:
12404:
12403:
12393:
12388:
12383:
12378:
12373:
12368:
12363:
12358:
12353:
12348:
12338:
12328:
12322:
12320:
12314:
12313:
12311:
12310:
12305:
12300:
12298:Lie derivative
12295:
12293:Integral curve
12290:
12285:
12280:
12279:
12278:
12268:
12263:
12262:
12261:
12254:Diffeomorphism
12251:
12245:
12243:
12237:
12236:
12234:
12233:
12228:
12223:
12218:
12213:
12208:
12203:
12198:
12193:
12187:
12185:
12176:
12175:
12173:
12172:
12167:
12162:
12157:
12152:
12147:
12142:
12137:
12132:
12131:
12130:
12125:
12115:
12114:
12113:
12102:
12100:
12099:Basic concepts
12096:
12095:
12083:
12082:
12075:
12068:
12060:
12054:
12053:
12048:
12034:
12033:External links
12031:
12029:
12028:
12016:
12003:
11978:
11973:978-0387782140
11972:
11948:
11934:
11913:
11907:
11891:
11877:
11856:
11850:
11837:
11825:(6): 338â341.
11801:
11782:
11776:
11760:
11754:
11738:
11732:
11719:
11717:Borel's review
11703:
11674:
11668:
11660:Academic Press
11651:
11646:978-3319134666
11645:
11624:
11618:
11593:
11571:
11539:
11533:
11513:
11493:
11487:
11474:
11439:
11426:
11404:
11394:
11377:
11363:
11346:
11344:
11341:
11338:
11337:
11326:
11324:, p. 131.
11314:
11302:
11290:
11258:
11246:
11234:
11222:
11210:
11198:
11186:
11174:
11162:
11149:
11137:
11125:
11107:
11095:
11092:Stillwell 2008
11084:
11073:
11058:
11056:Corollary 3.45
11046:
11034:
11022:
11020:, p. 100.
11010:
10998:
10965:
10953:
10941:
10926:
10900:
10899:
10897:
10894:
10891:
10890:
10881:
10867:
10866:
10864:
10861:
10859:
10856:
10854:
10853:
10847:
10842:
10837:
10832:
10827:
10822:
10816:
10814:
10811:
10810:
10809:
10786:
10759:
10752:
10694:locally convex
10675:
10672:
10671:
10670:
10663:tangent bundle
10651:parallelizable
10647:
10635:
10634:
10630:
10625:
10606:
10599:
10594:
10582:
10575:
10570:
10566:
10553:
10552:
10543:
10536:
10529:
10518:
10517:
10513:
10508:
10504:
10499:
10495:
10482:discrete group
10478:quotient group
10466:
10465:
10459:
10441:
10437:
10433:
10367:
10361:
10355:
10349:
10343:
10320:tangent spaces
10310:
10309:Classification
10307:
10251:
10231:
10228:
10225:
10222:
10216:
10213:
10186:
10162:
10139:
10136:
10109:
10106:
10103:
10100:
10094:
10091:
10058:Main article:
10055:
10052:
10039:
10019:
9999:
9968:
9948:
9924:
9921:
9918:
9915:
9912:
9908:
9905:
9884:
9881:
9877:
9873:
9870:
9835:
9815:
9792:
9768:
9729:
9709:
9685:
9661:
9641:
9626:
9623:
9575:
9574:
9562:
9559:
9556:
9553:
9550:
9545:
9541:
9537:
9534:
9531:
9528:
9525:
9522:
9519:
9516:
9513:
9510:
9507:
9504:
9479:
9474:
9471:
9449:
9444:
9439:
9434:
9429:
9425:
9404:
9401:
9398:
9395:
9392:
9370:
9367:
9364:
9361:
9358:
9355:
9352:
9349:
9346:
9343:
9340:
9337:
9334:
9331:
9328:
9325:
9322:
9319:
9299:
9279:
9268:
9267:
9256:
9252:
9248:
9245:
9242:
9239:
9236:
9233:
9230:
9227:
9224:
9221:
9217:
9211:
9208:
9202:
9199:
9196:
9193:
9190:
9187:
9184:
9181:
9178:
9174:
9168:
9165:
9159:
9156:
9153:
9150:
9147:
9144:
9138:
9135:
9129:
9126:
9123:
9120:
9116:
9112:
9109:
9106:
9103:
9100:
9097:
9094:
9091:
9087:
9084:
9081:
9078:
9075:
9052:
9049:
9046:
9043:
9040:
9018:
8996:
8965:
8945:
8925:
8905:
8881:
8861:
8838:
8834:
8830:
8827:
8824:
8820:
8817:
8796:
8792:
8788:
8785:
8782:
8779:
8754:
8728:
8707:
8687:
8665:
8648:diffeomorphism
8635:
8613:
8596:
8595:
8581:
8578:
8575:
8572:
8569:
8566:
8563:
8560:
8557:
8554:
8551:
8524:
8504:
8484:
8473:
8472:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8434:
8431:
8428:
8425:
8422:
8419:
8416:
8393:
8373:
8353:
8344:. Saying that
8333:
8330:
8327:
8324:
8321:
8317:
8314:
8293:
8290:
8286:
8282:
8279:
8259:
8239:
8217:
8195:
8168:
8148:
8128:
8108:
8104:
8100:
8097:
8094:
8090:
8087:
8066:
8046:
8035:
8034:
8023:
8020:
8014:
8011:
8005:
8001:
7995:
7989:
7986:
7980:
7976:
7970:
7967:
7964:
7961:
7958:
7955:
7952:
7949:
7946:
7943:
7916:
7912:
7908:
7905:
7902:
7898:
7895:
7874:
7870:
7866:
7863:
7860:
7856:
7853:
7829:
7825:
7821:
7818:
7815:
7811:
7781:Main article:
7778:
7775:
7751:
7748:
7745:
7695:
7694:
7682:
7660:
7656:
7635:
7632:
7627:
7623:
7602:
7599:
7596:
7593:
7590:
7570:
7548:
7543:
7538:
7533:
7530:
7508:
7484:
7462:
7442:
7415:
7395:
7380:is said to be
7369:
7347:
7344:
7282:
7279:
7266:
7246:
7235:diffeomorphism
7200:
7199:
7188:
7183:
7178:
7173:
7168:
7163:
7159:
7133:
7129:
7117:tangent spaces
7090:
7086:
7065:
7062:
7059:
7056:
7053:
7002:
6999:
6988:
6959:
6958:
6935:
6925:
6914:
6878:
6865:
6864:
6858:
6844:
6822:
6817:
6794:
6782:
6778:
6770:
6758:
6753:
6747:
6743:
6728:
6721:
6716:
6712:
6705:) =
6697:
6672:
6661:
6656:, because the
6630:
6629:
6617:
6614:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6590:
6587:
6555:
6552:
6547:
6540: in
6537:
6529:
6526:
6523:
6520:
6517:
6514:
6511:
6508:
6504:
6500:
6496:
6492:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6468:
6465:
6462:
6459:
6456:
6445:
6444:
6429:
6354:
6321:
6289:Main article:
6286:
6283:
6281:
6280:Basic concepts
6278:
6277:
6276:
6251:, such as the
6245:
6229:
6226:
6223:
6220:
6217:
6212:
6208:
6187:
6167:
6143:
6140:
6139:
6138:
6120:
6116:
6094:
6078:
6075:
6060:
6052:
6049:
6048:
6047:
6035:
6032:
6027:
6023:
6019:
5997:
5994:
5989:
5961:
5958:
5955:
5952:
5949:
5929:
5925:
5921:
5918:
5915:
5912:
5893:
5884:
5875:
5866:
5857:
5848:
5837:
5830:Poincaré group
5826:
5811:
5795:
5776:
5758:
5754:
5733:
5713:
5710:
5707:
5682:
5662:
5659:
5656:
5639:
5636:
5621:
5616:
5590:
5587:
5584:
5564:
5561:
5558:
5555:
5533:
5529:
5506:
5502:
5477:
5464:
5461:
5430:
5427:
5418:
5417:
5384:
5381:
5351:
5348:
5332:
5315:
5312:
5308:
5305:
5285:
5281:
5277:
5274:
5271:
5268:
5265:
5262:
5258:
5255:
5210:
5206:
5202:
5199:
5196:
5193:
5190:
5179:
5178:
5175:countably many
5168:
5156:
5152:
5148:
5145:
5142:
5139:
5136:
5112:
5108:
5104:
5101:
5098:
5095:
5092:
5052:
5048:
5044:
5041:
5038:
5035:
5032:
5004:
5001:
4896:
4871:
4867:
4863:
4860:
4857:
4854:
4851:
4829:
4826:
4818:
4817:
4805:
4801:
4797:
4794:
4791:
4788:
4785:
4765:
4762:
4759:
4756:
4753:
4750:
4728:
4725:
4721:
4717:
4711:
4706:
4685:
4682:
4679:
4659:
4655:
4651:
4648:
4645:
4642:
4639:
4619:
4615:
4611:
4608:
4605:
4602:
4587:
4575:
4572:
4569:
4566:
4563:
4543:
4540:
4537:
4534:
4531:
4528:
4506:
4503:
4499:
4495:
4490:
4486:
4465:
4462:
4459:
4439:
4435:
4431:
4428:
4425:
4422:
4419:
4399:
4395:
4391:
4388:
4385:
4382:
4371:unitary groups
4367:
4355:
4333:
4312:
4309:
4306:
4286:
4282:
4278:
4275:
4272:
4269:
4266:
4246:
4242:
4238:
4235:
4232:
4229:
4226:
4205:
4183:
4149:
4145:
4141:
4138:
4135:
4132:
4129:
4104:
4083:
4080:
4077:
4057:
4053:
4049:
4046:
4043:
4040:
4037:
4025:
4022:
4005:
3982:
3962:
3942:
3922:
3900:
3896:
3873:
3869:
3848:
3825:
3822:
3817:
3813:
3809:
3804:
3800:
3779:
3756:
3736:
3733:
3730:
3708:
3703:
3681:
3652:
3647:
3621:
3601:
3581:
3561:
3541:
3521:
3492:
3487:
3454:
3450:
3446:
3442:
3439:
3428:
3427:
3416:
3412:
3407:
3403:
3400:
3397:
3394:
3390:
3386:
3377:
3374:
3371:
3368:
3364:
3360:
3358:
3355:
3354:
3351:
3348:
3344:
3341:
3338:
3335:
3331:
3327:
3326:
3323:
3318:
3314:
3309:
3304:
3299:
3295:
3290:
3286:
3283:
3279:
3275:
3266:
3263:
3260:
3257:
3254:
3250:
3246:
3244:
3241:
3240:
3237:
3234:
3230:
3227:
3224:
3221:
3217:
3213:
3212:
3209:
3204:
3200:
3197:
3178:
3175:
3174:
3173:
3172:
3171:
3160:
3156:
3152:
3149:
3145:
3142:
3139:
3136:
3132:
3128:
3121:
3118:
3116:
3113:
3112:
3109:
3106:
3104:
3101:
3100:
3097:
3093:
3090:
3077:
3076:
3064:
3061:
3058:
3042:
3041:
3029:
3025:
3021:
3018:
3015:
3012:
3009:
2997:
2996:
2995:
2994:
2983:
2979:
2974:
2970:
2967:
2963:
2958:
2954:
2951:
2947:
2942:
2936:
2933:
2930:
2927:
2925:
2922:
2919:
2916:
2915:
2912:
2909:
2906:
2903:
2900:
2898:
2895:
2892:
2889:
2888:
2886:
2880:
2876:
2873:
2869:
2865:
2862:
2859:
2856:
2853:
2840:
2839:
2823:
2795:
2791:
2787:
2784:
2781:
2778:
2775:
2755:
2751:
2747:
2744:
2741:
2738:
2735:
2715:
2714:
2692:
2687:
2669:
2668:
2667:
2666:
2655:
2651:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2616:
2611:
2605:
2602:
2600:
2597:
2596:
2593:
2590:
2588:
2585:
2584:
2582:
2577:
2574:
2570:
2566:
2563:
2559:
2555:
2552:
2549:
2546:
2543:
2530:
2529:
2517:
2513:
2509:
2504:
2500:
2496:
2476:
2472:
2468:
2465:
2462:
2459:
2456:
2432:
2431:First examples
2429:
2421:
2420:
2409:
2404:
2401:
2397:
2393:
2390:
2387:
2384:
2381:
2378:
2343:
2342:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2267:multiplication
2255:real Lie group
2250:
2247:
2166:Poincaré group
2100:
2095:
2068:argued in his
2012:absolute value
1999:
1996:
1928:
1927:
1908:
1897:
1847:Henri Poincaré
1796:groupes de Lie
1775:
1772:
1734:
1712:
1689:
1686:
1683:
1661:
1657:
1653:
1648:
1621:
1617:
1613:
1608:
1581:
1482:
1481:
1479:
1478:
1471:
1464:
1456:
1453:
1452:
1449:
1448:
1446:Elliptic curve
1442:
1441:
1435:
1434:
1428:
1427:
1421:
1416:
1415:
1412:
1411:
1406:
1405:
1402:
1399:
1395:
1391:
1390:
1389:
1384:
1382:Diffeomorphism
1378:
1377:
1372:
1367:
1361:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1319:
1318:
1307:
1306:
1295:
1294:
1283:
1282:
1271:
1270:
1259:
1258:
1247:
1246:
1239:Special linear
1235:
1234:
1227:General linear
1223:
1222:
1217:
1211:
1202:
1201:
1198:
1197:
1194:
1193:
1188:
1183:
1175:
1174:
1161:
1149:
1136:
1123:
1121:Modular groups
1119:
1118:
1117:
1112:
1099:
1083:
1080:
1079:
1074:
1068:
1067:
1066:
1063:
1062:
1057:
1056:
1055:
1054:
1049:
1044:
1041:
1035:
1034:
1028:
1027:
1026:
1025:
1019:
1018:
1012:
1011:
1006:
997:
996:
994:Hall's theorem
991:
989:Sylow theorems
985:
984:
979:
971:
970:
969:
968:
962:
957:
954:Dihedral group
950:
949:
944:
938:
933:
927:
922:
911:
906:
905:
902:
901:
896:
895:
894:
893:
888:
880:
879:
878:
877:
872:
867:
862:
857:
852:
847:
845:multiplicative
842:
837:
832:
827:
819:
818:
817:
816:
811:
803:
802:
794:
793:
792:
791:
789:Wreath product
786:
781:
776:
774:direct product
768:
766:Quotient group
760:
759:
758:
757:
752:
747:
737:
734:
733:
730:
729:
721:
720:
700:
699:
697:
696:
689:
682:
674:
671:
670:
669:
668:
663:
655:
654:
650:
649:
644:
642:Harish-Chandra
639:
634:
629:
624:
619:
617:Henri Poincaré
614:
608:
607:
604:
603:
600:
599:
595:
594:
589:
584:
579:
573:
572:
567:Lie groups in
566:
565:
562:
561:
557:
556:
551:
546:
541:
536:
531:
525:
524:
519:
518:
515:
514:
510:
509:
504:
499:
497:
496:
491:
485:
483:
481:
480:
475:
469:
467:
462:
456:
455:
450:
449:
446:
445:
441:
440:
435:
430:
425:
423:
422:
417:
411:
409:
404:
399:
393:
392:
387:
386:
383:
382:
378:
377:
372:
367:
365:Diffeomorphism
362:
357:
352:
347:
341:
340:
335:
334:
331:
330:
325:
324:
323:
322:
318:
314:
310:
306:
302:
298:
294:
290:
286:
279:
278:
274:
273:
272:
271:
265:
261:
255:
251:
245:
241:
235:
228:
227:
221:
216:
215:
212:
211:
207:
206:
196:
186:
176:
166:
156:
149:Special linear
146:
139:General linear
135:
134:
129:
128:
125:
124:
116:
115:
101:
100:
55:
53:
46:
26:
9:
6:
4:
3:
2:
12965:
12954:
12951:
12949:
12946:
12944:
12941:
12940:
12938:
12923:
12920:
12918:
12917:Supermanifold
12915:
12913:
12910:
12908:
12905:
12901:
12898:
12897:
12896:
12893:
12891:
12888:
12886:
12883:
12881:
12878:
12876:
12873:
12871:
12868:
12866:
12863:
12862:
12860:
12856:
12850:
12847:
12845:
12842:
12840:
12837:
12835:
12832:
12830:
12827:
12825:
12822:
12821:
12819:
12815:
12805:
12802:
12800:
12797:
12795:
12792:
12790:
12787:
12785:
12782:
12780:
12777:
12775:
12772:
12770:
12767:
12765:
12762:
12760:
12757:
12756:
12754:
12752:
12748:
12742:
12739:
12737:
12734:
12732:
12729:
12727:
12724:
12722:
12719:
12717:
12714:
12712:
12708:
12704:
12702:
12699:
12697:
12694:
12692:
12688:
12684:
12682:
12679:
12677:
12674:
12672:
12669:
12667:
12664:
12662:
12659:
12657:
12654:
12653:
12651:
12649:
12645:
12639:
12638:Wedge product
12636:
12634:
12631:
12627:
12624:
12623:
12622:
12619:
12617:
12614:
12610:
12607:
12606:
12605:
12602:
12600:
12597:
12595:
12592:
12590:
12587:
12583:
12582:Vector-valued
12580:
12579:
12578:
12575:
12573:
12570:
12566:
12563:
12562:
12561:
12558:
12556:
12553:
12551:
12548:
12547:
12545:
12541:
12535:
12532:
12530:
12527:
12525:
12522:
12518:
12515:
12514:
12513:
12512:Tangent space
12510:
12508:
12505:
12503:
12500:
12498:
12495:
12494:
12492:
12488:
12485:
12483:
12479:
12473:
12470:
12468:
12464:
12460:
12458:
12455:
12453:
12449:
12445:
12441:
12439:
12436:
12434:
12431:
12429:
12426:
12424:
12421:
12419:
12416:
12414:
12411:
12409:
12406:
12402:
12399:
12398:
12397:
12394:
12392:
12389:
12387:
12384:
12382:
12379:
12377:
12374:
12372:
12369:
12367:
12364:
12362:
12359:
12357:
12354:
12352:
12349:
12347:
12343:
12339:
12337:
12333:
12329:
12327:
12324:
12323:
12321:
12315:
12309:
12306:
12304:
12301:
12299:
12296:
12294:
12291:
12289:
12286:
12284:
12281:
12277:
12276:in Lie theory
12274:
12273:
12272:
12269:
12267:
12264:
12260:
12257:
12256:
12255:
12252:
12250:
12247:
12246:
12244:
12242:
12238:
12232:
12229:
12227:
12224:
12222:
12219:
12217:
12214:
12212:
12209:
12207:
12204:
12202:
12199:
12197:
12194:
12192:
12189:
12188:
12186:
12183:
12179:Main results
12177:
12171:
12168:
12166:
12163:
12161:
12160:Tangent space
12158:
12156:
12153:
12151:
12148:
12146:
12143:
12141:
12138:
12136:
12133:
12129:
12126:
12124:
12121:
12120:
12119:
12116:
12112:
12109:
12108:
12107:
12104:
12103:
12101:
12097:
12092:
12088:
12081:
12076:
12074:
12069:
12067:
12062:
12061:
12058:
12052:
12049:
12046:
12041:
12037:
12036:
12022:
12017:
12014:
12010:
12006:
12000:
11996:
11992:
11988:
11984:
11979:
11975:
11969:
11965:
11961:
11957:
11953:
11949:
11945:
11941:
11937:
11931:
11927:
11923:
11919:
11914:
11910:
11904:
11900:
11896:
11892:
11888:
11884:
11880:
11874:
11870:
11866:
11862:
11857:
11853:
11847:
11843:
11838:
11833:
11828:
11824:
11820:
11819:
11814:
11812:
11806:
11802:
11798:
11794:
11793:
11788:
11783:
11779:
11777:4-00-006142-9
11773:
11769:
11765:
11761:
11757:
11751:
11747:
11743:
11739:
11735:
11729:
11725:
11720:
11718:
11714:
11710:
11706:
11700:
11695:
11690:
11686:
11682:
11681:
11675:
11671:
11669:0-12-329650-1
11665:
11661:
11657:
11652:
11648:
11642:
11638:
11634:
11630:
11625:
11621:
11615:
11611:
11607:
11603:
11599:
11594:
11590:
11586:
11582:
11578:
11574:
11568:
11564:
11560:
11556:
11552:
11548:
11544:
11540:
11536:
11530:
11526:
11522:
11518:
11514:
11510:
11506:
11502:
11498:
11494:
11490:
11484:
11480:
11475:
11473:
11472:3-540-43405-4
11469:
11465:
11464:3-540-42650-7
11461:
11457:
11456:3-540-64242-0
11453:
11448:
11444:
11440:
11437:
11433:
11429:
11423:
11419:
11415:
11414:
11409:
11408:Borel, Armand
11405:
11401:
11400:ScienceDirect
11397:
11391:
11386:
11385:
11378:
11374:
11370:
11366:
11360:
11356:
11352:
11348:
11347:
11335:
11330:
11323:
11322:Helgason 1978
11318:
11311:
11306:
11299:
11294:
11275:
11268:
11262:
11255:
11250:
11243:
11238:
11231:
11226:
11220:Corollary 5.7
11219:
11214:
11208:Section 1.3.4
11207:
11202:
11195:
11190:
11183:
11178:
11171:
11166:
11159:
11153:
11146:
11141:
11134:
11133:Helgason 1978
11129:
11118:
11111:
11104:
11099:
11093:
11088:
11082:
11081:Rossmann 2001
11077:
11070:
11065:
11063:
11055:
11050:
11043:
11042:Rossmann 2001
11038:
11031:
11026:
11019:
11014:
11008:, p. 43.
11007:
11002:
10993:
10988:
10984:
10980:
10976:
10969:
10963:, p. 76.
10962:
10957:
10950:
10945:
10938:
10933:
10931:
10915:
10911:
10905:
10901:
10885:
10878:
10872:
10868:
10851:
10848:
10846:
10843:
10841:
10838:
10836:
10833:
10831:
10828:
10826:
10823:
10821:
10818:
10817:
10807:
10803:
10799:
10795:
10791:
10787:
10784:
10780:
10776:
10772:
10768:
10764:
10760:
10757:
10753:
10750:
10746:
10742:
10738:
10734:
10730:
10726:
10723:The group of
10722:
10721:
10720:
10717:
10714:
10710:
10706:
10702:
10697:
10695:
10691:
10686:
10682:
10681:Banach spaces
10668:
10667:tangent space
10664:
10660:
10656:
10652:
10648:
10645:
10641:
10640:
10639:
10629:
10626:
10623:
10620:
10616:
10612:
10605:
10598:
10595:
10592:
10588:
10581:
10574:
10571:
10565:
10561:
10558:
10557:
10556:
10549:
10542:
10535:
10528:
10523:
10522:
10521:
10512:
10509:
10503:
10500:
10494:
10491:
10490:
10489:
10487:
10483:
10479:
10475:
10471:
10463:
10460:
10457:
10453:
10451:
10445:
10442:
10438:
10434:
10431:
10427:
10423:
10419:
10418:
10417:
10415:
10410:
10408:
10404:
10400:
10396:
10395:connectedness
10392:
10388:
10384:
10380:
10376:
10371:
10364:
10358:
10352:
10346:
10341:
10336:
10332:
10328:
10323:
10321:
10317:
10306:
10304:
10300:
10294:
10292:
10288:
10284:
10280:
10276:
10271:
10269:
10265:
10249:
10229:
10226:
10223:
10220:
10211:
10200:
10184:
10176:
10175:Hydrogen atom
10160:
10134:
10123:
10107:
10104:
10101:
10098:
10089:
10078:
10071:
10067:
10061:
10051:
10037:
10017:
9997:
9989:
9985:
9980:
9966:
9946:
9938:
9922:
9919:
9913:
9882:
9871:
9868:
9861:
9857:
9853:
9849:
9833:
9813:
9804:
9790:
9782:
9766:
9758:
9754:
9750:
9746:
9743:
9727:
9707:
9699:
9698:inclusion map
9683:
9675:
9659:
9639:
9632:
9622:
9620:
9619:Fréchet space
9617:
9616:
9610:
9608:
9601:
9596:
9594:
9586:
9582:
9580:
9560:
9551:
9543:
9539:
9532:
9529:
9526:
9517:
9511:
9508:
9502:
9495:
9494:
9493:
9472:
9469:
9432:
9427:
9423:
9402:
9396:
9393:
9390:
9381:
9365:
9362:
9359:
9353:
9350:
9347:
9341:
9335:
9332:
9326:
9320:
9317:
9297:
9277:
9254:
9250:
9246:
9243:
9237:
9234:
9228:
9225:
9222:
9209:
9206:
9200:
9194:
9191:
9185:
9182:
9179:
9166:
9163:
9157:
9151:
9148:
9145:
9136:
9133:
9127:
9124:
9121:
9118:
9114:
9110:
9107:
9104:
9098:
9092:
9089:
9082:
9076:
9073:
9066:
9065:
9064:
9050:
9047:
9044:
9041:
9038:
8994:
8986:
8982:
8977:
8963:
8943:
8936:generated by
8923:
8903:
8896:of the group
8895:
8879:
8859:
8850:
8828:
8825:
8786:
8783:
8777:
8769:
8743:
8705:
8685:
8654:of 0 in
8653:
8649:
8633:
8601:
8579:
8573:
8567:
8564:
8558:
8552:
8549:
8542:
8541:
8540:
8538:
8522:
8502:
8482:
8453:
8447:
8441:
8435:
8432:
8426:
8423:
8420:
8414:
8407:
8406:
8405:
8391:
8371:
8351:
8331:
8328:
8322:
8315:
8312:
8291:
8280:
8277:
8257:
8237:
8193:
8184:
8180:
8166:
8146:
8126:
8098:
8095:
8064:
8044:
8037:for matrices
8021:
8018:
8012:
8009:
8003:
7999:
7993:
7987:
7984:
7978:
7974:
7968:
7965:
7962:
7959:
7956:
7950:
7944:
7941:
7934:
7933:
7932:
7930:
7906:
7903:
7864:
7861:
7843:
7819:
7816:
7800:
7794:
7790:
7784:
7774:
7772:
7767:
7765:
7749:
7746:
7743:
7735:
7731:
7727:
7723:
7719:
7715:
7711:
7706:
7704:
7699:
7680:
7658:
7654:
7633:
7630:
7625:
7621:
7600:
7594:
7591:
7588:
7568:
7531:
7528:
7460:
7440:
7432:
7429:
7428:
7427:
7413:
7393:
7385:
7384:
7367:
7357:
7353:
7343:
7341:
7336:
7334:
7330:
7326:
7322:
7318:
7314:
7310:
7306:
7302:
7297:
7295:
7294:Ado's theorem
7291:
7286:
7278:
7264:
7244:
7236:
7232:
7228:
7223:
7221:
7217:
7213:
7209:
7205:
7186:
7166:
7161:
7157:
7149:
7148:
7147:
7131:
7127:
7118:
7114:
7110:
7106:
7088:
7084:
7063:
7057:
7054:
7051:
7043:
7038:
7036:
7032:
7028:
7024:
7020:
7016:
7012:
7008:
6998:
6996:
6992:
6984:
6980:
6976:
6972:
6968:
6964:
6957:
6954:
6950:
6946:
6942:
6941:
6940:
6938:
6930:
6928:
6921:
6917:
6909:
6906:
6902:
6898:
6894:
6861:
6820:
6806:
6802:
6798:
6791:
6790:tangent space
6787:
6781:
6775: =
6773:
6768:
6764:
6759:
6756:
6746:
6739:
6735:
6731:
6724:
6715:
6708:
6704:
6700:
6693:
6690: =
6689:
6685:
6681:
6677:
6673:
6670:
6666:
6662:
6659:
6655:
6651:
6647:
6644:
6640:
6636:
6635:
6634:
6615:
6612:
6609:
6606:
6603:
6600:
6594:
6591:
6588:
6577:
6573:
6569:
6553:
6535:
6527:
6524:
6518:
6515:
6509:
6506:
6490:
6487:
6481:
6478:
6475:
6469:
6463:
6457:
6454:
6447:
6446:
6442:
6438:
6434:
6430:
6427:
6424: +
6423:
6419:
6415:
6411:
6407:
6403:
6399:
6395:
6391:
6387:
6383:
6379:
6375:
6371:
6367:
6363:
6359:
6355:
6352:
6348:
6342:
6338:
6334:
6330:
6326:
6322:
6315:
6311:
6307:
6306:
6305:
6303:
6299:
6292:
6274:
6270:
6266:
6262:
6258:
6254:
6250:
6246:
6243:
6224:
6221:
6218:
6206:
6185:
6165:
6157:
6156:
6155:
6153:
6149:
6136:
6118:
6114:
6083:
6079:
6076:
6073:
6069:
6065:
6061:
6058:
6057:
6056:
6051:Constructions
6033:
6030:
6025:
6021:
6017:
5995:
5992:
5977:
5976:
5959:
5956:
5953:
5950:
5947:
5919:
5916:
5913:
5898:
5894:
5891:
5887:
5883:
5878:
5874:
5869:
5865:
5860:
5856:
5851:
5847:
5842:
5838:
5835:
5831:
5827:
5824:
5820:
5816:
5815:Lorentz group
5812:
5809:
5793:
5785:
5781:
5777:
5774:
5756:
5752:
5731:
5708:
5680:
5660:
5657:
5654:
5646:
5642:
5641:
5635:
5619:
5602:
5588:
5585:
5582:
5559:
5553:
5531:
5527:
5504:
5500:
5492:
5460:
5458:
5454:
5450:
5446:
5445:Matrix groups
5440:
5436:
5426:
5424:
5415:
5411:
5407:
5403:
5399:
5382:
5379:
5370:
5366:
5349:
5346:
5337:
5333:
5330:
5313:
5310:
5306:
5303:
5275:
5272:
5266:
5263:
5260:
5256:
5253:
5244:
5240:
5236:
5232:
5231:
5230:
5227:
5222:
5200:
5197:
5191:
5188:
5176:
5172:
5169:
5146:
5143:
5137:
5134:
5127:is closed in
5126:
5102:
5099:
5093:
5090:
5082:
5078:
5074:
5070:
5066:
5065:
5064:
5042:
5039:
5033:
5030:
5022:
5018:
5014:
5010:
5000:
4998:
4994:
4993:Lie groupoids
4990:
4986:
4982:
4978:
4973:
4971:
4967:
4963:
4959:
4955:
4951:
4947:
4943:
4939:
4935:
4931:
4929:
4925:
4921:
4920:-adic numbers
4919:
4914:
4912:
4885:
4861:
4858:
4852:
4849:
4841:
4837:
4836:
4825:
4823:
4795:
4792:
4786:
4783:
4763:
4760:
4754:
4726:
4723:
4719:
4715:
4704:
4683:
4680:
4677:
4649:
4646:
4640:
4637:
4609:
4606:
4600:
4592:
4588:
4570:
4564:
4561:
4541:
4538:
4532:
4504:
4501:
4497:
4493:
4488:
4484:
4463:
4460:
4457:
4429:
4426:
4420:
4417:
4389:
4386:
4380:
4372:
4368:
4310:
4307:
4304:
4276:
4273:
4267:
4264:
4236:
4233:
4227:
4224:
4172:
4168:
4167:
4166:
4163:
4139:
4136:
4130:
4127:
4119:
4081:
4078:
4075:
4047:
4044:
4038:
4035:
4021:
4019:
4003:
3994:
3980:
3960:
3940:
3920:
3898:
3894:
3871:
3867:
3846:
3839:
3838:in the group
3823:
3820:
3815:
3811:
3807:
3802:
3798:
3777:
3754:
3734:
3731:
3728:
3706:
3679:
3670:
3666:
3650:
3635:
3619:
3599:
3579:
3559:
3539:
3519:
3512:
3508:
3490:
3476:
3472:
3469:
3440:
3437:
3414:
3410:
3401:
3398:
3395:
3392:
3388:
3384:
3375:
3372:
3369:
3366:
3362:
3356:
3349:
3342:
3339:
3336:
3333:
3329:
3321:
3316:
3312:
3307:
3297:
3293:
3284:
3281:
3277:
3273:
3264:
3261:
3258:
3255:
3252:
3248:
3242:
3235:
3228:
3225:
3222:
3219:
3215:
3207:
3202:
3198:
3195:
3188:
3187:
3186:
3184:
3158:
3150:
3147:
3143:
3140:
3137:
3134:
3130:
3126:
3119:
3114:
3107:
3102:
3095:
3091:
3088:
3081:
3080:
3079:
3078:
3062:
3059:
3056:
3048:
3044:
3043:
3019:
3016:
3010:
3007:
2999:
2998:
2981:
2977:
2968:
2965:
2961:
2952:
2949:
2945:
2940:
2934:
2931:
2928:
2923:
2920:
2917:
2910:
2907:
2904:
2901:
2896:
2893:
2890:
2884:
2878:
2874:
2863:
2860:
2854:
2851:
2844:
2843:
2842:
2841:
2837:
2821:
2813:
2809:
2808:diffeomorphic
2785:
2782:
2776:
2773:
2766:, denoted by
2745:
2742:
2736:
2733:
2725:
2721:
2717:
2716:
2712:
2708:
2690:
2675:
2671:
2670:
2653:
2649:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2614:
2609:
2603:
2598:
2591:
2586:
2580:
2575:
2572:
2568:
2564:
2553:
2550:
2544:
2541:
2534:
2533:
2532:
2531:
2502:
2498:
2494:
2466:
2463:
2457:
2454:
2446:
2442:
2439:
2435:
2434:
2428:
2426:
2407:
2402:
2399:
2395:
2385:
2382:
2379:
2369:
2368:
2367:
2365:
2360:
2356:
2352:
2348:
2329:
2326:
2323:
2317:
2314:
2311:
2305:
2301:
2295:
2292:
2289:
2286:
2283:
2276:
2275:
2274:
2272:
2268:
2264:
2260:
2256:
2246:
2244:
2242:
2237:
2233:
2232:number theory
2229:
2225:
2221:
2217:
2213:
2209:
2205:
2201:
2200:Ellis Kolchin
2196:
2194:
2190:
2186:
2182:
2178:
2174:
2169:
2167:
2163:
2159:
2155:
2151:
2147:
2143:
2138:
2136:
2132:
2128:
2124:
2121:, whereas in
2120:
2116:
2098:
2083:
2079:
2075:
2071:
2067:
2063:
2058:
2056:
2052:
2048:
2044:
2040:
2036:
2033:
2025:
2021:
2020:complex plane
2017:
2013:
2009:
2004:
1995:
1993:
1989:
1983:
1981:
1977:
1976:Fifth Problem
1973:
1972:David Hilbert
1968:
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1925:
1921:
1917:
1913:
1909:
1906:
1902:
1898:
1895:
1891:
1890:
1889:
1887:
1882:
1880:
1876:
1872:
1868:
1864:
1860:
1856:
1855:Galois theory
1852:
1848:
1844:
1840:
1839:modular forms
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1804:
1799:
1797:
1793:
1789:
1785:
1780:
1771:
1769:
1765:
1764:Galois theory
1761:
1757:
1753:
1749:
1701:
1687:
1684:
1681:
1646:
1606:
1592:contained in
1579:
1571:
1566:
1564:
1560:
1556:
1551:
1549:
1545:
1540:
1536:
1532:
1527:
1525:
1521:
1517:
1516:
1508:
1493:
1489:
1477:
1472:
1470:
1465:
1463:
1458:
1457:
1455:
1454:
1447:
1444:
1443:
1440:
1437:
1436:
1433:
1430:
1429:
1426:
1423:
1422:
1419:
1414:
1413:
1403:
1400:
1397:
1396:
1394:
1388:
1385:
1383:
1380:
1379:
1376:
1373:
1371:
1368:
1366:
1363:
1362:
1359:
1353:
1351:
1345:
1343:
1337:
1335:
1329:
1327:
1321:
1320:
1316:
1312:
1309:
1308:
1304:
1300:
1297:
1296:
1292:
1288:
1285:
1284:
1280:
1276:
1273:
1272:
1268:
1264:
1261:
1260:
1256:
1252:
1249:
1248:
1244:
1240:
1237:
1236:
1232:
1228:
1225:
1224:
1221:
1218:
1216:
1213:
1212:
1209:
1205:
1200:
1199:
1192:
1189:
1187:
1184:
1182:
1179:
1178:
1150:
1125:
1124:
1122:
1116:
1113:
1088:
1085:
1084:
1078:
1075:
1073:
1070:
1069:
1065:
1064:
1053:
1050:
1048:
1045:
1042:
1039:
1038:
1037:
1036:
1033:
1030:
1029:
1024:
1021:
1020:
1017:
1014:
1013:
1010:
1007:
1005:
1003:
999:
998:
995:
992:
990:
987:
986:
983:
980:
978:
975:
974:
973:
972:
966:
963:
960:
955:
952:
951:
947:
942:
939:
936:
931:
928:
925:
920:
917:
916:
915:
914:
909:
908:Finite groups
904:
903:
892:
889:
887:
884:
883:
882:
881:
876:
873:
871:
868:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
822:
821:
820:
815:
812:
810:
807:
806:
805:
804:
801:
800:
796:
795:
790:
787:
785:
782:
780:
777:
775:
772:
769:
767:
764:
763:
762:
761:
756:
753:
751:
748:
746:
743:
742:
741:
740:
735:Basic notions
732:
731:
727:
723:
722:
719:
714:
710:
706:
705:
695:
690:
688:
683:
681:
676:
675:
673:
672:
667:
664:
662:
659:
658:
657:
656:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
609:
602:
601:
593:
590:
588:
585:
583:
580:
578:
575:
574:
570:
564:
563:
555:
552:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
526:
522:
517:
516:
508:
505:
503:
500:
495:
492:
490:
487:
486:
484:
479:
476:
474:
471:
470:
468:
466:
463:
461:
458:
457:
453:
448:
447:
439:
436:
434:
431:
429:
426:
421:
418:
416:
413:
412:
410:
408:
405:
403:
400:
398:
395:
394:
390:
385:
384:
376:
373:
371:
368:
366:
363:
361:
358:
356:
353:
351:
348:
346:
343:
342:
338:
333:
332:
321:
315:
313:
307:
305:
299:
297:
291:
289:
283:
282:
281:
280:
276:
275:
270:
268:
262:
260:
258:
252:
250:
248:
242:
240:
238:
232:
231:
230:
229:
225:
224:
219:
214:
213:
204:
200:
197:
194:
190:
187:
184:
180:
177:
174:
170:
167:
164:
160:
157:
154:
150:
147:
144:
140:
137:
136:
132:
127:
126:
122:
118:
117:
114:
110:
107:
106:
97:
94:
86:
76:
72:
66:
65:
59:
54:
45:
44:
39:
35:
30:
19:
12844:Moving frame
12839:Morse theory
12829:Gauge theory
12621:Tensor field
12550:Closed/Exact
12529:Vector field
12497:Distribution
12438:Hypercomplex
12433:Quaternionic
12395:
12170:Vector field
12128:Smooth atlas
11982:
11955:
11926:10.1142/6515
11917:
11898:
11860:
11841:
11822:
11816:
11810:
11796:
11790:
11767:
11745:
11723:
11679:
11655:
11628:
11597:
11550:
11520:
11500:
11478:
11446:
11412:
11398:– via
11383:
11354:
11329:
11317:
11305:
11300:Theorem 5.20
11293:
11281:. Retrieved
11274:the original
11261:
11256:Theorem 3.42
11249:
11244:Section 13.2
11237:
11225:
11213:
11201:
11196:Example 3.27
11189:
11184:Theorem 5.20
11177:
11165:
11152:
11147:Theorem 3.20
11140:
11128:
11110:
11098:
11087:
11076:
11049:
11044:, Chapter 2.
11037:
11025:
11018:Hawkins 2000
11013:
11006:Hawkins 2000
11001:
10982:
10978:
10968:
10961:Hawkins 2000
10956:
10951:, p. 2.
10949:Hawkins 2000
10944:
10939:, p. 1.
10937:Hawkins 2000
10917:. Retrieved
10913:
10904:
10884:
10871:
10825:Haar measure
10805:
10801:
10789:
10729:Witt algebra
10718:
10712:
10708:
10704:
10700:
10698:
10677:
10661:between its
10657:(there is a
10636:
10627:
10621:
10619:circle group
10614:
10603:
10596:
10579:
10572:
10563:
10559:
10554:
10547:
10540:
10533:
10526:
10519:
10510:
10501:
10492:
10485:
10467:
10449:
10425:
10411:
10405:) and their
10372:
10362:
10356:
10350:
10344:
10324:
10312:
10295:
10283:Hermann Weyl
10278:
10274:
10272:
10198:
10073:
9981:
9860:homomorphism
9855:
9851:
9805:
9631:Lie subgroup
9630:
9628:
9625:Lie subgroup
9614:
9606:
9597:
9590:
9576:
9382:
9269:
8980:
8978:
8893:
8851:
8652:neighborhood
8599:
8597:
8474:
8185:
8181:
8036:
7796:
7768:
7722:integer spin
7707:
7702:
7696:
7430:
7381:
7360:A Lie group
7359:
7337:
7320:
7316:
7312:
7308:
7304:
7300:
7298:
7287:
7284:
7226:
7224:
7201:
7112:
7108:
7039:
7025:is a smooth
7022:
7018:
7014:
7010:
7006:
7004:
6986:
6978:
6974:
6970:
6966:
6962:
6960:
6955:
6952:
6948:
6944:
6933:
6931:
6923:
6919:
6912:
6910:
6904:
6900:
6896:
6892:
6866:
6859:
6800:
6792:
6785:
6776:
6771:
6766:
6762:
6751:
6741:
6737:
6733:
6726:
6719:
6710:
6706:
6702:
6695:
6691:
6687:
6683:
6679:
6675:
6668:
6664:
6653:
6649:
6645:
6638:
6631:
6575:
6567:
6440:
6436:
6432:
6425:
6421:
6417:
6413:
6409:
6405:
6401:
6397:
6393:
6389:
6385:
6381:
6377:
6373:
6369:
6365:
6361:
6357:
6350:
6346:
6340:
6336:
6332:
6328:
6313:
6309:
6294:
6272:
6268:
6260:
6256:
6253:Galois group
6241:
6147:
6145:
6134:
6071:
6054:
5973:
5881:
5872:
5863:
5854:
5845:
5603:
5491:circle group
5466:
5442:
5422:
5419:
5413:
5410:power series
5405:
5401:
5364:
5335:
5328:
5234:
5225:
5223:
5180:
5173:has at most
5170:
5124:
5080:
5076:
5072:
5068:
5020:
5016:
5006:
4995:, which are
4981:group object
4974:
4953:
4949:
4932:
4927:
4923:
4917:
4910:
4909:
4833:
4831:
4819:
4161:
4027:
4018:Lie subgroup
3995:
3837:
3769:
3636:subgroup of
3511:neighborhood
3467:
3429:
3180:
2836:parametrized
2707:disconnected
2424:
2422:
2363:
2358:
2354:
2346:
2344:
2254:
2252:
2240:
2204:Armand Borel
2197:
2170:
2139:
2134:
2059:
2050:
2046:
2029:
2024:circle group
1987:
1984:
1969:
1957:Hermann Weyl
1940:
1936:
1929:
1883:
1830:
1814:
1800:
1795:
1791:
1777:
1567:
1559:circle group
1552:
1528:
1494:(pronounced
1491:
1485:
1314:
1302:
1290:
1278:
1266:
1254:
1242:
1230:
1207:
1001:
958:
945:
934:
923:
919:Cyclic group
797:
784:Free product
755:Group action
718:Group theory
713:Group theory
712:
647:Armand Borel
632:Hermann Weyl
433:Loop algebra
415:Killing form
389:Lie algebras
266:
256:
246:
236:
202:
192:
182:
172:
162:
152:
142:
113:Lie algebras
108:
89:
80:
61:
29:
12789:Levi-Civita
12779:Generalized
12751:Connections
12701:Lie algebra
12633:Volume form
12534:Vector flow
12507:Pushforward
12502:Lie bracket
12401:Lie algebra
12366:G-structure
12155:Pushforward
12135:Submanifold
11547:Harris, Joe
11232:Theorem 5.6
10779:loop groups
10763:gauge group
10713:Lie algebra
10569:is discrete
10436:dimensions.
10407:compactness
9755:, a closed
7212:Lie bracket
7115:with their
6995:Lie bracket
6899:determines
6658:Lie bracket
6643:derivations
5645:group SU(2)
3532:of a point
3183:uncountable
3177:Non-example
2838:as follows:
2711:determinant
2345:means that
2271:smooth maps
2236:adele rings
2146:Lie algebra
2131:G-structure
2066:Felix Klein
2055:Lie algebra
1945:Ălie Cartan
1907:and Jacobi;
1875:quadratures
1843:Felix Klein
1784:Felix Klein
1488:mathematics
1204:Topological
1043:alternating
627:Ălie Cartan
473:Root system
277:Exceptional
75:introducing
12943:Lie groups
12937:Categories
12912:Stratifold
12870:Diffeology
12666:Associated
12467:Symplectic
12452:Riemannian
12381:Hyperbolic
12308:Submersion
12216:HopfâRinow
12150:Submersion
12145:Smooth map
12045:Lie groups
11811:Lie groups
11501:Lie Groups
11343:References
11283:11 October
11030:Borel 2001
10914:aimath.org
10476:, and the
10462:Semisimple
10456:classified
10420:Connected
10379:semisimple
10327:direct sum
10264:classified
10064:See also:
8650:between a
8404:and that
8304:such that
7787:See also:
7613:such that
7433:: Suppose
7350:See also:
7227:isomorphic
7208:linear map
7105:derivative
6732:for every
6686:acting on
6566:where exp(
6302:commutator
6244:manifolds.
5819:isometries
5453:orthogonal
5433:See also:
5296:such that
5063:such that
4942:Montgomery
4741:(and also
4519:(and also
3996:The group
3770:The group
2674:noncompact
2177:Riemannian
1982:in Paris.
1779:Sophus Lie
1752:Sophus Lie
1674:groups of
1572:subgroups
1311:Symplectic
1251:Orthogonal
1208:Lie groups
1115:Free group
840:continuous
779:Direct sum
612:Sophus Lie
605:Scientists
478:Weyl group
199:Symplectic
159:Orthogonal
109:Lie groups
58:references
12948:Manifolds
12794:Principal
12769:Ehresmann
12726:Subbundle
12716:Principal
12691:Fibration
12671:Cotangent
12543:Covectors
12396:Lie group
12376:Hermitian
12319:manifolds
12288:Immersion
12283:Foliation
12221:Noether's
12206:Frobenius
12201:De Rham's
12196:Darboux's
12087:Manifolds
11809:"Review:
11589:246650103
11310:Hall 2015
11298:Hall 2015
11254:Hall 2015
11242:Hall 2015
11230:Hall 2015
11218:Hall 2015
11206:Hall 2015
11194:Hall 2015
11182:Hall 2015
11170:Hall 2015
11158:Hall 2015
11145:Hall 2015
11069:Hall 2015
11054:Hall 2015
10896:Citations
10705:Lie group
10399:connected
10393:), their
10387:nilpotent
10230:ψ
10221:ψ
10215:^
10185:ψ
10161:ψ
10138:^
10108:ψ
10099:ψ
10093:^
9914:φ
9880:→
9869:φ
9745:immersion
9742:injective
9544:∗
9540:ϕ
9533:
9512:
9503:ϕ
9473:∈
9443:→
9428:∗
9424:ϕ
9400:→
9391:ϕ
9354:
9336:
9321:
9247:⋯
9244:−
9201:−
9111:
9093:
9077:
9048:∈
8770:(because
8553:
8289:→
8022:⋯
7945:
7747:≥
7681:ϕ
7659:∗
7655:ϕ
7626:∗
7622:ϕ
7598:→
7589:ϕ
7542:→
7521:and that
7231:bijective
7177:→
7167::
7162:∗
7158:ϕ
7132:∗
7128:ϕ
7089:∗
7085:ϕ
7061:→
7055::
7052:ϕ
6610:−
6525:∈
6510:
6479:∈
6458:
6211:∞
6135:universal
5954:×
5843:of types
5784:nilpotent
5658:×
5586:×
5267:
5261:⊂
5226:Lie group
5192:
5138:
5094:
5034:
5009:Hausdorff
4915:over the
4853:
4724:−
4681:×
4502:−
4489:∗
4461:×
4308:×
4079:×
3941:θ
3821:∈
3732:∈
3449:∖
3441:∈
3402:∈
3399:ϕ
3393:θ
3376:ϕ
3370:π
3343:θ
3337:π
3298:⊂
3285:∈
3282:θ
3265:θ
3256:π
3229:θ
3223:π
3151:∈
3060:×
2969:π
2953:∈
2950:φ
2935:φ
2932:
2924:φ
2921:
2911:φ
2908:
2902:−
2897:φ
2894:
2822:φ
2643:≠
2634:−
2400:−
2392:↦
2306:μ
2299:→
2293:×
2284:μ
2074:invariant
1924:Grassmann
1815:idée fixe
1685:×
1492:Lie group
1375:Conformal
1263:Euclidean
870:nilpotent
489:Real form
375:Euclidean
226:Classical
83:June 2023
38:Ree group
12953:Symmetry
12890:Orbifold
12885:K-theory
12875:Diffiety
12599:Pullback
12413:Oriented
12391:Kenmotsu
12371:Hadamard
12317:Types of
12266:Geodesic
12091:Glossary
11954:(2008).
11897:(1965),
11807:(1959).
11744:(2002),
11549:(1991).
11519:(2003).
11499:(1957).
11410:(2001),
11353:(1969),
11312:Part III
11156:But see
10985:: 1â88.
10813:See also
10798:M-theory
10758:gravity.
10756:quantize
10731:, whose
10617:and the
10383:solvable
10301:and the
9986:gives a
9781:embedded
9757:subgroup
9579:commutes
9492:we have
9063:we have
8768:matrices
8475:for all
8316:′
7646:, where
7042:category
7035:analytic
7017: :
6740:, where
6312:is just
6154:), are:
5744:-sphere
5383:′
5350:′
5314:′
5257:′
4985:category
3859:joining
2724:subgroup
2720:rotation
2436:The 2Ă2
2189:analysis
2164:and the
2156:(or its
2133:, where
2062:geometry
1998:Overview
1970:In 1900
1912:geometry
1863:symmetry
1531:manifold
1370:Poincaré
1215:Solenoid
1087:Integers
1077:Lattices
1052:sporadic
1047:Lie type
875:solvable
865:dihedral
850:additive
835:infinite
745:Subgroup
661:Glossary
355:Poincaré
12834:History
12817:Related
12731:Tangent
12709:)
12689:)
12656:Adjoint
12648:Bundles
12626:density
12524:Torsion
12490:Vectors
12482:Tensors
12465:)
12450:)
12446:,
12444:Pseudoâ
12423:Poisson
12356:Finsler
12351:Fibered
12346:Contact
12344:)
12336:Complex
12334:)
12303:Section
12013:0722297
11944:2382250
11887:0835009
11713:1771134
11581:1153249
11436:1847105
11373:0252560
10919:1 March
10790:simpler
10391:abelian
9937:closure
7840:of the
7734:Spin(n)
7431:Theorem
7220:functor
7103:be its
6973:,
6969:sends (
6965:×
6805:Fraktur
6725:=
6682:, with
5821:of the
5724:is the
5224:Then a
4983:in the
4938:Gleason
3692:inside
2810:to the
2076:. Thus
1916:PlĂŒcker
1905:Poisson
1774:History
1563:physics
1518:) is a
1365:Lorentz
1287:Unitary
1186:Lattice
1126:PSL(2,
860:abelian
771:(Semi-)
569:physics
350:Lorentz
179:Unitary
71:improve
12799:Vector
12784:Koszul
12764:Cartan
12759:Affine
12741:Vector
12736:Tensor
12721:Spinor
12711:Normal
12707:Stable
12661:Affine
12565:bundle
12517:bundle
12463:Almost
12386:KĂ€hler
12342:Almost
12332:Almost
12326:Closed
12226:Sard's
12182:(list)
12011:
12001:
11970:
11942:
11932:
11905:
11885:
11875:
11848:
11774:
11752:
11730:
11711:
11701:
11666:
11643:
11616:
11587:
11579:
11569:
11531:
11509:529830
11507:
11485:
11470:
11462:
11454:
11434:
11424:
11392:
11371:
11361:
10448:SL(2,
10375:simple
10335:simple
10329:of an
10068:, and
9935:. The
9740:is an
9674:subset
9605:SL(2,
8583:
8460:
7703:unique
7301:global
6435:of GL(
6388:is in
5834:affine
4946:Zippin
4116:. Any
2812:circle
2487:or by
2206:, and
2047:global
2032:smooth
2016:circle
1955:, and
1920:Möbius
1877:, the
1672:, the
1570:matrix
1548:smooth
1220:Circle
1151:SL(2,
1040:cyclic
1004:-group
855:cyclic
830:finite
825:simple
809:kernel
345:Circle
60:, but
12907:Sheaf
12681:Fiber
12457:Rizza
12428:Prime
12259:Local
12249:Curve
12111:Atlas
12024:(PDF)
11277:(PDF)
11270:(PDF)
11120:(PDF)
10858:Notes
10796:. In
10739:(see
10585:is a
10555:Then
10480:is a
10360:and D
10199:space
9895:with
9846:be a
9700:from
8139:into
8057:. If
7764:Sp(n)
7730:SU(n)
7710:SU(2)
7333:SO(3)
7329:SU(2)
7311:then
6977:) to
6404:with
6376:of M(
6247:Some
5396:is a
4173:over
3634:dense
3475:torus
3468:fixed
3430:with
2362:into
2259:group
2257:is a
2243:-adic
2216:field
2179:or a
2051:local
2010:with
1894:group
1702:over
1520:group
1404:Sp(â)
1401:SU(â)
814:image
420:Index
12774:Form
12676:Dual
12609:flow
12472:Tame
12448:Subâ
12361:Flat
12241:Maps
11999:ISBN
11968:ISBN
11930:ISBN
11903:ISBN
11873:ISBN
11846:ISBN
11772:ISBN
11750:ISBN
11728:ISBN
11699:ISBN
11664:ISBN
11641:ISBN
11614:ISBN
11585:OCLC
11567:ISBN
11529:ISBN
11505:OCLC
11483:ISBN
11468:ISBN
11460:ISBN
11452:ISBN
11422:ISBN
11390:ISBN
11359:ISBN
11285:2014
10921:2024
10773:and
10735:the
10642:The
10525:1 â
10468:The
9982:The
9747:and
9600:onto
9290:and
8495:and
7797:The
7791:and
7736:for
7724:and
7497:and
7453:and
7354:and
7331:and
7315:and
7111:and
7009:and
6951:) â
6080:The
6062:Any
5895:The
5839:The
5828:The
5813:The
5778:The
5643:The
5455:and
5437:and
5123:and
4944:and
4630:and
4589:The
4410:and
4369:The
4257:and
4195:and
4169:The
4028:Let
3138:>
3045:The
2718:The
2438:real
2173:acts
2160:),
2082:E(3)
1857:and
1845:and
1825:and
1490:, a
1398:O(â)
1387:Loop
1206:and
370:Loop
111:and
12696:Jet
11991:doi
11960:doi
11922:doi
11865:doi
11827:doi
11689:doi
11633:doi
11606:doi
11559:doi
10987:doi
10711:in
10709:Lie
10703:in
10701:Lie
10631:nil
10607:nil
10600:sol
10583:sol
10576:con
10567:con
10544:con
10537:sol
10530:nil
10514:nil
10505:sol
10496:con
10401:or
10354:, C
10348:, B
10316:Lie
9939:of
9850:of
9759:of
9720:to
9676:of
9530:exp
9509:exp
9351:exp
9333:exp
9318:exp
9108:exp
9090:exp
9074:exp
8872:of
8698:in
8550:exp
8230:of
7942:exp
7885:to
7005:If
6985:on
6961:on
6953:xyx
6736:in
6663:If
6507:exp
6455:Lie
6356:If
6327:GL(
6267:to
6148:not
6070:or
5978:on
5447:or
5075:in
4886:of
4749:det
4527:det
4345:or
4120:of
3886:to
3592:in
3552:in
2929:cos
2918:sin
2905:sin
2891:cos
2726:of
2619:det
1724:or
1632:or
1514:LEE
1486:In
1313:Sp(
1301:SU(
1277:SO(
1241:SL(
1229:GL(
201:Sp(
191:SU(
171:SO(
151:SL(
141:GL(
36:or
12939::
12687:Co
12009:MR
12007:,
11997:,
11989:,
11966:.
11940:MR
11938:,
11928:,
11883:MR
11881:.
11871:.
11823:65
11821:.
11815:.
11795:,
11789:,
11709:MR
11707:,
11697:,
11687:,
11662:.
11658:.
11639:,
11612:.
11604:.
11600:.
11583:.
11577:MR
11575:.
11565:.
11553:.
11545:;
11523:.
11445:,
11432:MR
11430:,
11420:,
11369:MR
11367:,
11061:^
10983:18
10981:.
10977:.
10929:^
10912:.
10546:â
10539:â
10532:â
10432:).
10409:.
10389:,
10385:,
10381:,
10377:,
10305:.
10293:.
10285:.
9979:.
9629:A
9581:,
9210:12
9167:12
8976:.
7773:.
7766:.
7037:.
7021:â
6947:,
6929:.
6729:gh
6707:gh
6654:YX
6650:XY
6568:tX
6439:,
6406:AA
6396:,
6380:,
6364:,
6351:BA
6347:AB
6339:,
6331:,
6198:,
5907:Sp
5879:,
5870:,
5861:,
5852:,
5702:SU
5371:,
5264:GL
5189:GL
5135:GL
5091:GL
5031:GL
5011:)
4972:.
4940:,
4850:SL
4832:A
4824:.
4217:,
3665:.
3466:a
2427:.
2357:Ă
2253:A
2238:;
2202:,
2195:.
2168:.
2113:,
2057:.
1994:.
1967:.
1922:,
1918:,
1770:.
1642:GL
1602:GL
1565:.
1542:a
1529:A
1504:iË
1289:U(
1265:E(
1253:O(
711:â
181:U(
161:O(
12705:(
12685:(
12461:(
12442:(
12340:(
12330:(
12093:)
12089:(
12079:e
12072:t
12065:v
11993::
11976:.
11962::
11947:.
11924::
11912:.
11889:.
11867::
11835:.
11829::
11797:1
11781:.
11759:.
11736:.
11691::
11672:.
11650:.
11635::
11622:.
11608::
11591:.
11561::
11537:.
11511:.
11492:.
11402:.
11376:.
11287:.
11071:.
11032:.
10995:.
10989::
10923:.
10806:N
10802:N
10785:.
10628:G
10624:.
10622:S
10615:R
10604:G
10602:/
10597:G
10593:.
10580:G
10578:/
10573:G
10564:G
10562:/
10560:G
10551:.
10548:G
10541:G
10534:G
10527:G
10511:G
10502:G
10493:G
10486:G
10452:)
10450:R
10426:S
10397:(
10368:8
10363:n
10357:n
10351:n
10345:n
10279:K
10275:K
10250:E
10227:E
10224:=
10212:H
10135:H
10105:E
10102:=
10090:H
10038:G
10018:G
9998:G
9967:G
9947:H
9923:H
9920:=
9917:)
9911:(
9907:m
9904:i
9883:G
9876:R
9872::
9856:G
9834:H
9814:G
9791:G
9767:G
9728:G
9708:H
9684:G
9660:G
9640:H
9615:C
9609:)
9607:R
9561:.
9558:)
9555:)
9552:x
9549:(
9536:(
9527:=
9524:)
9521:)
9518:x
9515:(
9506:(
9478:g
9470:x
9448:h
9438:g
9433::
9403:H
9397:G
9394::
9369:)
9366:Y
9363:+
9360:X
9357:(
9348:=
9345:)
9342:Y
9339:(
9330:)
9327:X
9324:(
9298:Y
9278:X
9255:,
9251:)
9241:]
9238:X
9235:,
9232:]
9229:Y
9226:,
9223:X
9220:[
9216:[
9207:1
9198:]
9195:Y
9192:,
9189:]
9186:Y
9183:,
9180:X
9177:[
9173:[
9164:1
9158:+
9155:]
9152:Y
9149:,
9146:X
9143:[
9137:2
9134:1
9128:+
9125:Y
9122:+
9119:X
9115:(
9105:=
9102:)
9099:Y
9096:(
9086:)
9083:X
9080:(
9051:U
9045:Y
9042:,
9039:X
9017:g
8995:U
8964:G
8944:N
8924:G
8904:G
8880:e
8860:N
8837:)
8833:R
8829:,
8826:n
8823:(
8819:L
8816:G
8795:)
8791:R
8787:,
8784:n
8781:(
8778:M
8753:C
8727:R
8706:G
8686:e
8664:g
8634:G
8612:g
8580:.
8577:)
8574:1
8571:(
8568:c
8565:=
8562:)
8559:X
8556:(
8523:G
8503:t
8483:s
8457:)
8454:t
8451:(
8448:c
8445:)
8442:s
8439:(
8436:c
8433:=
8430:)
8427:t
8424:+
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8418:(
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