Lie group - Knowledge

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

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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

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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

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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

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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

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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

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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

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considered the winter of 1873–1874 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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,

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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.

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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.

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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

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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.

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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,

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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

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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

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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

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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,

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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 ε is an infinitesimal positive number with ε = 0 (of course, no such real number ε 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.

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There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have

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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

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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

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that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties

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can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space

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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: 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

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or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its

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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

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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

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where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are

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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

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is in terms of the "highest weight" of the representation. The classification is closely related to the

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winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a

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Methods for determining whether a Lie group is simply connected or not are discussed in the article on

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Lie groups play an important role, via their connections with Galois representations in number theory.

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Bäuerle, 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

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was to develop a theory of symmetries of differential equations that would accomplish for them what

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can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write

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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

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is simple according to the second definition but not according to the first. They have all been

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homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a

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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

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that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.

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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

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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

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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

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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

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close" to the identity, and the Lie bracket of the Lie algebra is related to the

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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: Birkhäuser, 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:—i.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:(1842–1899), 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:

Die Zusammensetzung der stetigen endlichen Transformationsgruppen

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 ε = 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:Bäuerle, 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 group–Lie 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 group–Lie algebra correspondence 5239:Lie group–Lie 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 group–Lie 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 + ε 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:Hilbert–Smith 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:Borel–Weil–Bott 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 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 + ε 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. 304–317. 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:Atiyah–Singer 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 + ε 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:: 19–57, 152–193 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: 18:Matrix Lie group 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:Évariste 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): 338–341. 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:Élie Cartan 1907:and Jacobi; 1875:quadratures 1843:Felix Klein 1784:Felix Klein 1488:mathematics 1204:Topological 1043:alternating 627:Élie 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:Hopf–Rinow 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:: 1–88. 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 2×2 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:Pseudo− 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:Plücker 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:Kähler 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(∞) 1401:SU(∞) 814:image 420:Index 12774:Form 12676:Dual 12609:flow 12472:Tame 12448:Sub− 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 ⊆ 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:) → 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(∞) 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:⊆ 10539:⊆ 10532:⊆ 10432:). 10409:. 10389:, 10385:, 10381:, 10377:, 10305:. 10293:. 10285:. 9979:. 9629:A 9581:, 9210:12 9167:12 8976:. 7773:. 7766:. 7037:. 7021:→ 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:× 2253:A 2238:; 2202:, 2195:. 2168:. 2113:, 2057:. 1994:. 1967:. 1922:, 1918:, 1770:. 1642:GL 1602:GL 1565:. 1542:a 1529:A 1504:iː 1289:U( 1265:E( 1253:O( 711:→ 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:+ 8421:s 8418:( 8415:c 8392:G 8372:c 8352:c 8332:X 8329:= 8326:) 8323:0 8320:( 8313:c 8292:G 8285:R 8281:: 8278:c 8258:G 8238:G 8216:g 8194:X 8167:G 8147:G 8127:G 8107:) 8103:C 8099:; 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