2567:, which is the maximum dimension of a subspace of the tangent space (to any point) on which the curvature is identically zero. The rank is always at least one, with equality if the sectional curvature is positive or negative. If the curvature is positive, the space is of compact type, and if negative, it is of noncompact type. The spaces of Euclidean type have rank equal to their dimension and are isometric to a Euclidean space of that dimension. Therefore, it remains to classify the irreducible, simply connected Riemannian symmetric spaces of compact and non-compact type. In both cases there are two classes.
44:
9257:
10426:
10076:
2512:
if it is not the product of two or more
Riemannian symmetric spaces. It can then be shown that any simply connected Riemannian symmetric space is a Riemannian product of irreducible ones. Therefore, we may further restrict ourselves to classifying the irreducible, simply connected Riemannian
9080:
1032:
Locally
Riemannian symmetric spaces that are not Riemannian symmetric may be constructed as quotients of Riemannian symmetric spaces by discrete groups of isometries with no fixed points, and as open subsets of (locally) Riemannian symmetric spaces.
10297:
9874:
10510:
on the tangent space at the identity coset. Thus the
Hermitian symmetric spaces are easily read off of the classification. In both the compact and the non-compact cases it turns out that there are four infinite series, namely AIII, BDI with
9400:
1582:
10170:
6091:
The classification of
Riemannian symmetric spaces does not extend readily to the general case for the simple reason that there is no general splitting of a symmetric space into a product of irreducibles. Here a symmetric space
10563:. Thus the quaternion-Kähler symmetric spaces are easily read off from the classification. In both the compact and the non-compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI with
9737:, as corresponds to the transpose for the orthogonal groups and the Hermitian conjugate for the unitary groups. It is a linear functional, and it is self-adjoint, and so one concludes that there is an orthonormal basis
10491:. Some examples are complex vector spaces and complex projective spaces, both with their usual Riemannian metric, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric.
2326:
6082:
is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the
Riemannian and pseudo-Riemannian case.
6269:
is semisimple. This is the analogue of the
Riemannian dichotomy between Euclidean spaces and those of compact or noncompact type, and it motivated M. Berger to classify semisimple symmetric spaces (i.e., those with
9252:{\displaystyle \langle X,Y\rangle _{\mathfrak {g}}={\begin{cases}\langle X,Y\rangle _{p}\quad &X,Y\in T_{p}M\cong {\mathfrak {m}}\\-B(X,Y)\quad &X,Y\in {\mathfrak {h}}\\0&{\mbox{otherwise}}\end{cases}}}
6145:
3251:
1449:
2052:
9475:
5256:
686:
In geometric terms, a complete, simply connected
Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold
4886:
4495:
4013:
3461:
2159:. Conversely, symmetric spaces with compact isotropy group are Riemannian symmetric spaces, although not necessarily in a unique way. To obtain a Riemannian symmetric space structure we need to fix a
5585:
4305:
5975:
5495:
5449:
5312:
5155:
5109:
4942:
4669:
4552:
4405:
4359:
1256:
9541:
9299:
2718:
Specializing to the
Riemannian symmetric spaces of class A and compact type, Cartan found that there are the following seven infinite series and twelve exceptional Riemannian symmetric spaces
4972:
9637:
10696:, in Representation Theory and Automorphic Forms: Instructional Conference, International Centre for Mathematical Sciences, March 1996, Edinburgh, Scotland, American Mathematical Society,
9672:
5395:
5055:
9863:
4788:
4207:
922:
962:
if its geodesic symmetries are in fact isometric. This is equivalent to the vanishing of the covariant derivative of the curvature tensor. A locally symmetric space is said to be a
2710:
In both class A and class B there is thus a correspondence between symmetric spaces of compact type and non-compact type. This is known as duality for
Riemannian symmetric spaces.
10518:, DIII and CI, and two exceptional spaces, namely EIII and EVII. The non-compact Hermitian symmetric spaces can be realized as bounded symmetric domains in complex vector spaces.
5928:
5733:
3957:
3405:
3190:
2138:
9781:
6668:
6610:
6575:
6512:
6422:
3609:
2994:
10421:{\displaystyle \langle \cdot ,\cdot \rangle ={\frac {1}{\lambda _{1}}}\left.B\right|_{{\mathfrak {m}}_{1}}+\cdots +{\frac {1}{\lambda _{d}}}\left.B\right|_{{\mathfrak {m}}_{d}}}
5845:
4120:
3788:
3565:
3358:
3030:
2835:
2799:
10071:{\displaystyle \langle Y_{i}^{\#},Y_{j}\rangle =\lambda _{i}\langle Y_{i},Y_{j}\rangle =B(Y_{i},Y_{j})=\langle Y_{j}^{\#},Y_{i}\rangle =\lambda _{j}\langle Y_{j},Y_{i}\rangle }
9434:
5775:
5657:
5625:
4701:
3821:
3747:
3642:
3142:
9731:
3909:
2949:
2140:
see the definition and following proposition on page 209, chapter IV, section 3 in
Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces for further information.
10289:
10103:
9805:
9696:
9574:
9507:
6536:
6446:
6391:
6363:
6316:
6292:
6267:
6224:
6200:
6172:
1860:
1836:
1808:
1781:
1757:
1729:
1705:
1681:
1657:
1633:
1609:
1398:
1370:
1338:
1302:
2411:, and then performs these two constructions in sequence, then the Riemannian symmetric space yielded is isometric to the original one. This shows that the "algebraic data" (
5903:
5878:
2891:
863:
3081:
2094:
6071:
4076:
3521:
3314:
10265:
5809:
5691:
5529:
5346:
5189:
5006:
4830:
4735:
4615:
4586:
4439:
4249:
4154:
9329:
3857:
3678:
2687:
is its maximal compact subgroup. Each such example has a corresponding example of compact type, by considering a maximal compact subgroup of the complexification of
1108:
739:
9334:
4040:
3278:
3108:
2918:
1460:
786:
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. Their central role in the theory of holonomy was discovered by
10236:
10111:
3485:
10469:
of a Riemannian manifold at a point acts irreducibly on the tangent space, then either the manifold is a locally Riemannian symmetric space, or it is in one of
3878:
6698:
The following table indexes the real symmetric spaces by complex symmetric spaces and real forms, for each classical and exceptional complex simple Lie group.
3713:
745:, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of
9034:
An account of weakly symmetric spaces and their classification by Akhiezer and Vinberg, based on the classification of periodic automorphisms of complex
10435:
the spectrum of the hydrogen atom, with the eigenvalues of the Killing form corresponding to different values of the angular momentum of an orbital (
2679:
The examples of class A are completely described by the classification of noncompact simply connected real simple Lie groups. For non-compact type,
461:
1735:, but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature of symmetric spaces is the third condition that
5923:
construction. The irreducible compact Riemannian symmetric spaces are, up to finite covers, either a compact simple Lie group, a Grassmannian, a
6627:
The classification therefore reduces to the classification of commuting pairs of antilinear involutions of a complex Lie algebra. The composite
10487:
A Riemannian symmetric space that is additionally equipped with a parallel complex structure compatible with the Riemannian metric is called a
2581:
is either the product of a compact simple Lie group with itself (compact type), or a complexification of such a Lie group (non-compact type).
509:
8953:
Selberg's definition can also be phrased equivalently in terms of a generalization of geodesic symmetry. It is required that for every point
6106:
3196:
1410:
2480:
of a Riemannian symmetric space is again Riemannian symmetric, and the covering map is described by dividing the connected isometry group
514:
2695:. More directly, the examples of compact type are classified by involutive automorphisms of compact simply connected simple Lie groups
2238:
1068:
of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space.
504:
499:
1987:
10993:(1956), "Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces, with applications to Dirichlet series",
9439:
319:
6294:
semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if
5195:
583:
466:
4836:
4445:
3963:
3411:
5535:
4255:
11024:
10838:
10701:
5934:
5454:
5408:
5271:
5114:
5068:
4901:
4628:
4511:
4364:
4318:
2734:, together with a geometric interpretation, if readily available. The labelling of these spaces is the one given by Cartan.
10917:
979:
614:
10537:
10527:
1192:
790:. They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.
9509:
can be further factored into eigenspaces classified by the Killing form. This is accomplished by defining an adjoint map
1280:
fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra
9512:
9265:
2493:
4947:
1947:. This action is faithful (e.g., by a theorem of Kostant, any isometry in the identity component is determined by its
1110:, which is symmetric. The lens spaces are quotients of the 3-sphere by a discrete isometry that has no fixed points.
10927:
10879:
10856:
10773:
10751:
9582:
10889:
10622:
9645:
5352:
5012:
22:
9813:
4741:
4160:
2639:
476:
7700:
to be the identity involution (indicated by a dash). In the above tables this is implicitly covered by the case
868:
11042:
6341:. Any semisimple symmetric space is a product of symmetric spaces of this form with symmetric spaces such that
10536:) isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is called
471:
451:
11047:
10870:
Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions
416:
324:
5697:
3921:
3369:
3154:
2099:
11057:
10470:
9740:
6642:
6584:
6549:
6486:
6396:
3576:
2961:
1732:
5815:
4090:
3758:
3535:
3328:
3000:
2805:
2769:
10431:
In certain practical applications, this factorization can be interpreted as the spectrum of operators,
9409:
6175:
5748:
5630:
5598:
5266:
4896:
4674:
3794:
3720:
3615:
3115:
2484:
of the covering by a subgroup of its center. Therefore, we may suppose without loss of generality that
1401:
746:
640:
456:
9701:
3885:
2925:
1587:
The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer
10560:
10488:
10482:
10270:
10084:
9786:
9677:
9546:
9488:
6517:
6427:
6372:
6344:
6297:
6273:
6248:
6205:
6181:
6153:
2440:
1841:
1817:
1789:
1762:
1738:
1710:
1686:
1662:
1638:
1614:
1590:
1379:
1351:
1319:
1283:
9116:
10602:
10596:
10466:
6639:
determines a real form. From this it is easy to construct tables of symmetric spaces for any given
5998:
5920:
2058:
1154:
772:
607:
91:
10446:
Classification of symmetric spaces proceeds based on whether or not the Killing form is definite.
5886:
5861:
2841:
1057:, each with their standard Riemannian metrics. More examples are provided by compact, semi-simple
833:
7696:
For exceptional simple Lie groups, the Riemannian case is included explicitly below, by allowing
6040:
Symmetric and locally symmetric spaces in general can be regarded as affine symmetric spaces. If
5924:
5743:
4506:
3036:
2072:
2626:
The compact simply connected Lie groups are the universal covers of the classical Lie groups SO(
9035:
8935:
8849:
6365:
is simple. It remains to describe the latter case. For this, one needs to classify involutions
4046:
3491:
3284:
411:
374:
342:
329:
10532:
A Riemannian symmetric space that is additionally equipped with a parallel subbundle of End(T
10244:
9395:{\displaystyle B(X,Y)=\operatorname {trace} (\operatorname {ad} X\circ \operatorname {ad} Y)}
6227:
5786:
5668:
5506:
5323:
5166:
4983:
4807:
4712:
4592:
4563:
4416:
4226:
4131:
676:
672:
443:
111:
9304:
5919:) uniformly classifies the Riemannian symmetric spaces, both compact and non-compact, via a
3827:
3648:
1577:{\displaystyle \subset {\mathfrak {h}},\;\subset {\mathfrak {m}},\;\subset {\mathfrak {h}}.}
1078:
714:
186:
176:
166:
156:
10890:"A uniform description of compact symmetric spaces as Grassmannians using the magic square"
10627:
10443:
that can classify the different representations under which different orbitals transform.)
10165:{\displaystyle {\mathfrak {m}}={\mathfrak {m}}_{1}\oplus \cdots \oplus {\mathfrak {m}}_{d}}
6026:
5985:. A similar construction produces the irreducible non-compact Riemannian symmetric spaces.
2516:
The next step is to show that any irreducible, simply connected Riemannian symmetric space
1114:
991:
71:
61:
10656:
Jurgen Jost, (2002) "Riemannian Geometry and Geometric Analysis", Third edition, Springer
8:
11052:
4019:
3257:
3087:
2897:
2617:
2546:
1865:
656:
644:
636:
600:
588:
429:
259:
10291:
semisimple, so that the Killing form is non-degenerate, the metric likewise factorizes:
10181:
6029:(with zero, positive and negative curvature respectively). De Sitter space of dimension
5993:
An important class of symmetric spaces generalizing the Riemannian symmetric spaces are
3467:
10979:
10868:
10740:
6075:
6014:
3863:
2699:(up to conjugation). Such involutions extend to involutions of the complexification of
2472:) be the algebraic data associated to it. To classify the possible isometry classes of
652:
360:
350:
6678:. This extends the compact/non-compact duality from the Riemannian case, where either
6448:
is a complex simple Lie algebra, and the corresponding symmetric spaces have the form
3684:
1021:). In fact, already the identity component of the isometry group acts transitively on
11020:
10923:
10875:
10852:
10834:
10769:
10747:
10697:
10637:
6687:
6059:
1010:
779:
This definition includes more than the Riemannian definition, and reduces to it when
742:
680:
424:
387:
539:
277:
10971:
10959:
10904:
10816:
10794:
10722:
10681:
10610:
10506:
contains a central circle. A quarter turn by this circle acts as multiplication by
10440:
6033:
may be identified with the 1-sheeted hyperboloid in a Minkowski space of dimension
6001:(nondegenerate instead of positive definite on each tangent space). In particular,
2585:
1054:
1050:
995:
944:
need not be isometric, nor can it be extended, in general, from a neighbourhood of
559:
239:
231:
223:
215:
207:
140:
121:
81:
10694:
Harmonic analysis on semisimple symmetric spaces: A survey of some general results
1810:
with a direct sum decomposition satisfying these three conditions, the linear map
10556:
6079:
6022:
6018:
2532:
2477:
1065:
1042:
544:
297:
282:
53:
10955:
10669:
10632:
10460:
8927:
6234:
6078:. Conversely a manifold with such a connection is locally symmetric (i.e., its
6063:
1915:
999:
987:
966:
if in addition its geodesic symmetries can be extended to isometries on all of
564:
382:
287:
10908:
10782:
2446:
668:
549:
11036:
10710:
9059:
933:
787:
272:
101:
10807:
Cartan, Élie (1927), "Sur une classe remarquable d'espaces de Riemann, II",
10990:
10761:
10735:
9403:
9071:
8931:
8855:
4080:
3525:
3318:
2062:
569:
554:
355:
337:
267:
9406:. The minus sign appears because the Killing form is negative-definite on
2584:
The examples in class B are completely described by the classification of
10606:
6233:
However, the irreducible symmetric spaces can be classified. As shown by
628:
395:
311:
35:
10727:
752:
From the point of view of Lie theory, a symmetric space is the quotient
10983:
10933:
Chapter XI contains a good introduction to Riemannian symmetric spaces.
10821:
10799:
10691:
10685:
6539:
5881:
1948:
1866:
Riemannian symmetric spaces satisfy the Lie-theoretic characterization
1072:
1058:
664:
534:
400:
292:
671:
to give a complete classification. Symmetric spaces commonly occur in
6329:
As in the Riemannian case there are semisimple symmetric spaces with
6140:{\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}}
3246:{\displaystyle \mathrm {S} (\mathrm {U} (p)\times \mathrm {U} (q))\,}
1883:
1444:{\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}}
1130:
1002:
locally Riemannian symmetric space is actually Riemannian symmetric.
761:
31:
10975:
1153:
of a typical point is an open subgroup of the fixed point set of an
10692:
van den Ban, E. P.; Flensted-Jensen, M.; Schlichtkrull, H. (1997),
6670:, and furthermore, there is an obvious duality given by exchanging
5856:
1312:, whose square is the identity. It follows that the eigenvalues of
660:
648:
2445:
The algebraic description of Riemannian symmetric spaces enabled
2321:{\displaystyle s_{p}:M\to M,\quad h'K\mapsto h\sigma (h^{-1}h')K}
491:
2616:
is its maximal compact subgroup. In both cases, the rank is the
2434:
10551:
is quaternion-Kähler if and only if isotropy representation of
2559:
has nonpositive (but not identically zero) sectional curvature.
2047:{\displaystyle \sigma :G\to G,h\mapsto s_{p}\circ h\circ s_{p}}
1046:
10785:(1926), "Sur une classe remarquable d'espaces de Riemann, I",
6226:
is not semisimple (or even reductive) in general, it can have
1075:
is locally symmetric but not symmetric, with the exception of
9470:{\displaystyle \langle \cdot ,\cdot \rangle _{\mathfrak {g}}}
6690:, i.e., its fixed point set is a maximal compact subalgebra.
6464:: these are the analogues of the Riemannian symmetric spaces
6009:
dimensional pseudo-Riemannian symmetric spaces of signature (
771:
that is (a connected component of) the invariant group of an
9050:
Some properties and forms of symmetric spaces can be noted.
8930:.) Selberg proved that weakly symmetric spaces give rise to
5251:{\displaystyle \mathrm {Spin} (16)/\{\pm \mathrm {vol} \}\,}
2508:
A simply connected Riemannian symmetric space is said to be
1894:
is Riemannian homogeneous). Therefore, if we fix some point
10833:, CBMS Regional Conference, American Mathematical Society,
10672:(1999), "Weakly symmetric spaces and spherical varieties",
10392:
10338:
9245:
8858:
extended Cartan's definition of symmetric space to that of
43:
18:(pseudo-)Riemannian manifold whose geodesics are reversible
2175:: such an inner product always exists by averaging, since
9868:
These are orthogonal with respect to the metric, in that
6054:
is a symmetric space, then Nomizu showed that there is a
4881:{\displaystyle \mathrm {SO} (12)\cdot \mathrm {SU} (2)\,}
4490:{\displaystyle \mathrm {SO} (10)\cdot \mathrm {SO} (2)\,}
4008:{\displaystyle \mathrm {Sp} (p)\times \mathrm {Sp} (q)\,}
3456:{\displaystyle \mathrm {SO} (p)\times \mathrm {SO} (q)\,}
2531:
has vanishing curvature, and is therefore isometric to a
695:) is said to be symmetric if and only if, for each point
655:
about every point. This can be studied with the tools of
10766:
Essays in the History of Lie Groups and Algebraic Groups
5580:{\displaystyle \mathrm {Sp} (3)\cdot \mathrm {SU} (2)\,}
4300:{\displaystyle \mathrm {SU} (6)\cdot \mathrm {SU} (2)\,}
1874:
is a Riemannian symmetric space, the identity component
6237:, there is a dichotomy: an irreducible symmetric space
5970:{\displaystyle (\mathbf {A} \otimes \mathbf {B} )^{n},}
5490:{\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}
5444:{\displaystyle (\mathbb {O} \otimes \mathbb {O} )P^{2}}
5307:{\displaystyle (\mathbb {O} \otimes \mathbb {O} )P^{2}}
5150:{\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}
5104:{\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}
4937:{\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}
4664:{\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}
4547:{\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}
4400:{\displaystyle (\mathbb {C} \otimes \mathbb {H} )P^{2}}
4354:{\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}
2703:, and these in turn classify non-compact real forms of
10948:
Symmetric spaces II: Compact Spaces and Classification
10849:
Differential geometry, Lie groups and symmetric spaces
10081:
since the Killing form is symmetric. This factorizes
9733:
is the Killing form. This map is sometimes called the
9236:
10521:
10300:
10273:
10247:
10184:
10114:
10087:
9877:
9816:
9789:
9743:
9704:
9680:
9648:
9585:
9549:
9515:
9491:
9442:
9412:
9337:
9307:
9268:
9083:
8843:
6645:
6587:
6552:
6520:
6489:
6430:
6399:
6375:
6347:
6300:
6276:
6251:
6208:
6184:
6156:
6109:
5937:
5889:
5864:
5818:
5789:
5751:
5700:
5671:
5633:
5601:
5538:
5509:
5457:
5411:
5355:
5326:
5274:
5198:
5169:
5117:
5071:
5015:
4986:
4950:
4904:
4839:
4810:
4744:
4715:
4677:
4631:
4595:
4566:
4514:
4448:
4419:
4367:
4321:
4258:
4229:
4163:
4134:
4093:
4049:
4022:
3966:
3924:
3888:
3866:
3830:
3797:
3761:
3723:
3687:
3651:
3618:
3579:
3538:
3494:
3470:
3414:
3372:
3331:
3287:
3260:
3199:
3157:
3118:
3090:
3039:
3003:
2964:
2928:
2900:
2844:
2808:
2772:
2449:
to obtain a complete classification of them in 1926.
2241:
2102:
2075:
1990:
1844:
1820:
1792:
1765:
1741:
1713:
1689:
1665:
1641:
1617:
1593:
1463:
1413:
1382:
1354:
1322:
1286:
1195:
1081:
871:
836:
717:
10960:"Invariant affine connections on homogeneous spaces"
8870:
with a transitive connected Lie group of isometries
2096:
and its identity component (hence an open subgroup)
1251:{\displaystyle G^{\sigma }=\{g\in G:\sigma (g)=g\}.}
10715:
Annales Scientifiques de l'École Normale Supérieure
2612:is a simply connected complex simple Lie group and
826:and reverses geodesics through that point, i.e. if
10867:
10739:
10613:can be interpreted as reductive symmetric spaces.
10420:
10283:
10259:
10230:
10164:
10097:
10070:
9857:
9799:
9775:
9725:
9690:
9666:
9631:
9568:
9536:{\displaystyle {\mathfrak {m}}\to {\mathfrak {m}}}
9535:
9501:
9469:
9428:
9394:
9323:
9293:
9251:
6662:
6604:
6569:
6538:may be viewed as the fixed point set of a complex
6530:
6506:
6440:
6416:
6385:
6357:
6310:
6286:
6261:
6218:
6194:
6166:
6139:
5997:, in which the Riemannian metric is replaced by a
5969:
5897:
5872:
5839:
5803:
5769:
5727:
5685:
5651:
5619:
5579:
5523:
5489:
5443:
5389:
5340:
5306:
5250:
5183:
5149:
5103:
5049:
5000:
4966:
4936:
4880:
4824:
4782:
4729:
4695:
4663:
4609:
4580:
4546:
4489:
4433:
4399:
4353:
4299:
4243:
4201:
4148:
4114:
4070:
4034:
4007:
3951:
3903:
3872:
3851:
3815:
3782:
3741:
3707:
3672:
3636:
3603:
3559:
3515:
3479:
3455:
3399:
3352:
3308:
3272:
3245:
3184:
3136:
3102:
3075:
3024:
2988:
2943:
2912:
2885:
2829:
2793:
2320:
2132:
2088:
2046:
1854:
1830:
1802:
1775:
1751:
1723:
1699:
1675:
1651:
1627:
1603:
1576:
1443:
1392:
1364:
1332:
1296:
1250:
1113:An example of a non-Riemannian symmetric space is
1102:
1041:Basic examples of Riemannian symmetric spaces are
916:
857:
733:
10862:The standard book on Riemannian symmetric spaces.
10571: = 2 (these are isomorphic), BDI with
10449:
9294:{\displaystyle \langle \cdot ,\cdot \rangle _{p}}
2163:-invariant inner product on the tangent space to
11034:
10757:Contains a compact introduction and many tables.
10454:
4967:{\displaystyle \mathbb {H} \otimes \mathbb {O} }
4050:
3495:
3288:
2608:is the diagonal subgroup. For non-compact type,
2592:is a compact simply connected simple Lie group,
2407:If one starts with a Riemannian symmetric space
1269:(including, of course, the identity component).
1061:equipped with a bi-invariant Riemannian metric.
741:as minus the identity (every symmetric space is
462:Representation theory of semisimple Lie algebras
10919:Foundations of Differential Geometry, Volume II
10915:
10888:Huang, Yongdong; Leung, Naichung Conan (2010).
10828:
10713:(1957), "Les espaces symétriques noncompacts",
10667:
9632:{\displaystyle \langle X,Y^{\#}\rangle =B(X,Y)}
10916:Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
10587: = 1, EII, EVI, EIX, FI and G.
6581:extends to a complex antilinear involution of
10809:Bulletin de la Société Mathématique de France
10787:Bulletin de la Société Mathématique de France
10476:
9667:{\displaystyle \langle \cdot ,\cdot \rangle }
5390:{\displaystyle E_{7}\cdot \mathrm {SU} (2)\,}
5050:{\displaystyle E_{6}\cdot \mathrm {SO} (2)\,}
2951:that leave the complex determinant invariant
2435:Classification of Riemannian symmetric spaces
1316:are ±1. The +1 eigenspace is the Lie algebra
608:
10313:
10301:
10065:
10039:
10023:
9992:
9951:
9925:
9909:
9878:
9858:{\displaystyle Y_{i}^{\#}=\lambda _{i}Y_{i}}
9661:
9649:
9605:
9586:
9456:
9443:
9282:
9269:
9132:
9119:
9097:
9084:
9053:
8866:. These are defined as Riemannian manifolds
6635:determines a complex symmetric space, while
5244:
5227:
4783:{\displaystyle \mathrm {SU} (8)/\{\pm I\}\,}
4776:
4767:
4202:{\displaystyle \mathrm {Sp} (4)/\{\pm I\}\,}
4195:
4186:
2210:is Riemannian symmetric, consider any point
1242:
1209:
10831:Analysis on Non-Riemannian Symmetric Spaces
10590:
6616:and hence also a complex linear involution
6230:representations which are not irreducible.
5988:
2545:has nonnegative (but not identically zero)
2069:is contained between the fixed point group
1964:, hence compact. Moreover, if we denote by
659:, leading to consequences in the theory of
10887:
10555:contains an Sp(1) summand acting like the
6245:is either flat (i.e., an affine space) or
6013: − 1,1), are important in
5916:
3717:Space of orthogonal complex structures on
1955:is a subgroup of the orthogonal group of T
1537:
1500:
924:It follows that the derivative of the map
917:{\displaystyle f(\gamma (t))=\gamma (-t).}
615:
601:
500:Particle physics and representation theory
42:
10820:
10798:
10726:
6086:
5891:
5866:
5836:
5800:
5753:
5724:
5682:
5635:
5603:
5576:
5520:
5470:
5462:
5424:
5416:
5386:
5337:
5287:
5279:
5247:
5180:
5130:
5122:
5084:
5076:
5046:
4997:
4960:
4952:
4917:
4909:
4877:
4821:
4779:
4726:
4683:
4680:
4644:
4636:
4606:
4577:
4527:
4519:
4486:
4430:
4380:
4372:
4334:
4326:
4296:
4240:
4198:
4145:
4096:
4004:
3948:
3891:
3812:
3779:
3726:
3633:
3600:
3541:
3452:
3396:
3334:
3242:
3181:
3121:
3021:
2985:
2931:
2826:
2790:
2126:
1348:), and the −1 eigenspace will be denoted
1186:is an open subgroup of the invariant set
10865:
10846:
2713:
2520:is of one of the following three types:
2503:
1265:is open, it is a union of components of
980:Cartan–Ambrose–Hicks theorem
11017:Harmonic Analysis on Commutative Spaces
10989:
2452:For a given Riemannian symmetric space
2427:) completely describe the structure of
1120:
802:be a connected Riemannian manifold and
793:
467:Representations of classical Lie groups
11035:
10954:
10806:
10781:
10709:
10760:
10734:
9066:can be lifted to a scalar product on
8926:was later shown to be unnecessary by
3144:compatible with the Hermitian metric
11014:
11005:
10945:
10936:
9301:is the Riemannian metric defined on
9039:
8323: / SO(9,1)×SO(1,1)
8132: / SO(5,5)×SO(1,1)
5728:{\displaystyle \mathrm {Spin} (9)\,}
3952:{\displaystyle \mathrm {Sp} (p+q)\,}
3400:{\displaystyle \mathrm {SO} (p+q)\,}
3185:{\displaystyle \mathrm {SU} (p+q)\,}
3112:Space of quaternionic structures on
2133:{\displaystyle (G^{\sigma })_{o}\,,}
1013:(meaning that the isometry group of
940:. On a general Riemannian manifold,
320:Lie group–Lie algebra correspondence
10405:
10351:
10276:
10208:
10191:
10151:
10128:
10117:
10090:
9792:
9776:{\displaystyle Y_{1},\ldots ,Y_{n}}
9683:
9528:
9518:
9494:
9461:
9415:
9221:
9174:
9102:
6663:{\displaystyle {\mathfrak {g}}^{c}}
6649:
6605:{\displaystyle {\mathfrak {g}}^{c}}
6591:
6570:{\displaystyle {\mathfrak {g}}^{c}}
6556:
6523:
6507:{\displaystyle {\mathfrak {g}}^{c}}
6493:
6433:
6417:{\displaystyle {\mathfrak {g}}^{c}}
6403:
6378:
6350:
6303:
6279:
6254:
6211:
6187:
6159:
6132:
6122:
6112:
5910:
3604:{\displaystyle \mathrm {SO} (2n)\,}
2989:{\displaystyle \mathrm {SU} (2n)\,}
2488:is simply connected. (This implies
1930:. By differentiating the action at
1847:
1823:
1795:
1768:
1744:
1716:
1692:
1668:
1644:
1620:
1596:
1566:
1553:
1543:
1529:
1516:
1506:
1492:
1479:
1469:
1436:
1426:
1416:
1385:
1357:
1325:
1289:
973:
13:
10939:Symmetric spaces I: General Theory
10922:, Wiley Classics Library edition,
10522:Quaternion-Kähler symmetric spaces
10005:
9891:
9827:
9600:
9561:
8844:Weakly symmetric Riemannian spaces
8622: / SO(10,2)×SL(2,
8544: / SO(8,4)×SU(2)
8524: / SO(12)× Sp(1)
8414: / SO(12)× Sp(1)
8264: / SO(8,2)×SO(2)
8247: / SU(4,2)×SU(2)
8198: / SO(6,4)×SO(2)
8181: / SU(4,2)×SU(2)
7951: / Sp(2,1)×Sp(1)
7914: / Sp(2,1)×Sp(1)
6017:, the most notable examples being
5995:pseudo-Riemannian symmetric spaces
5840:{\displaystyle \mathrm {SO} (4)\,}
5823:
5820:
5711:
5708:
5705:
5702:
5563:
5560:
5543:
5540:
5373:
5370:
5240:
5237:
5234:
5209:
5206:
5203:
5200:
5033:
5030:
4864:
4861:
4844:
4841:
4749:
4746:
4473:
4470:
4453:
4450:
4283:
4280:
4263:
4260:
4168:
4165:
4115:{\displaystyle \mathbb {H} ^{p+q}}
3991:
3988:
3971:
3968:
3929:
3926:
3911:compatible with the inner product
3799:
3783:{\displaystyle \mathrm {Sp} (n)\,}
3766:
3763:
3620:
3584:
3581:
3560:{\displaystyle \mathbb {R} ^{p+q}}
3439:
3436:
3419:
3416:
3377:
3374:
3353:{\displaystyle \mathbb {C} ^{p+q}}
3226:
3209:
3201:
3162:
3159:
3025:{\displaystyle \mathrm {Sp} (n)\,}
3008:
3005:
2969:
2966:
2830:{\displaystyle \mathrm {SO} (n)\,}
2813:
2810:
2794:{\displaystyle \mathrm {SU} (n)\,}
2777:
2774:
2726:. They are here given in terms of
2494:long exact sequence of a fibration
2396:is a geodesic symmetry and, since
1786:Conversely, given any Lie algebra
1635:. The second condition means that
1344:(since this is the Lie algebra of
932:is minus the identity map on the
14:
11069:
11019:, American Mathematical Society,
10768:, American Mathematical Society,
10538:quaternion-Kähler symmetric space
10528:Quaternion-Kähler symmetric space
10465:If the identity component of the
9429:{\displaystyle {\mathfrak {h}}~;}
8860:weakly symmetric Riemannian space
8474: / SO(6,6)×SL(2,
8253: / SU(5,1)×SL(2,
8233: / SO(10)×SO(2)
8187: / SU(3,3)×SL(2,
8058: / SO(10)×SO(2)
6062:(i.e. an affine connection whose
5770:{\displaystyle \mathbb {O} P^{2}}
5652:{\displaystyle \mathbb {H} P^{2}}
5620:{\displaystyle \mathbb {O} P^{2}}
4696:{\displaystyle \mathbb {OP} ^{2}}
3816:{\displaystyle \mathrm {U} (n)\,}
3742:{\displaystyle \mathbb {R} ^{2n}}
3637:{\displaystyle \mathrm {U} (n)\,}
3137:{\displaystyle \mathbb {C} ^{2n}}
2404:is a Riemannian symmetric space.
1973:: M → M the geodesic symmetry of
1934:we obtain an isometric action of
1906:is diffeomorphic to the quotient
1862:, is an involutive automorphism.
1005:Every Riemannian symmetric space
10829:Flensted-Jensen, Mogens (1986),
10623:Orthogonal symmetric Lie algebra
9726:{\displaystyle B(\cdot ,\cdot )}
9480:
8157: / SU(6)×SU(2)
8051: / SU(6)×SU(2)
7891: / Sp(3)×Sp(1)
7863: / Sp(3)×Sp(1)
7779: / SU(2)×SU(2)
7760: / SU(2)×SU(2)
6476:a complex simple Lie group, and
5950:
5942:
5595:Space of symmetric subspaces of
5405:Space of symmetric subspaces of
5065:Space of symmetric subspaces of
4625:Space of symmetric subspaces of
4315:Space of symmetric subspaces of
3904:{\displaystyle \mathbb {H} ^{n}}
2944:{\displaystyle \mathbb {C} ^{n}}
2563:A more refined invariant is the
2378:equal to minus the identity on T
1731:. Thus any symmetric space is a
986:is locally Riemannian symmetric
711:and acting on the tangent space
23:Symmetric space (disambiguation)
10543:An irreducible symmetric space
10494:An irreducible symmetric space
10284:{\displaystyle {\mathfrak {g}}}
10098:{\displaystyle {\mathfrak {m}}}
9800:{\displaystyle {\mathfrak {m}}}
9691:{\displaystyle {\mathfrak {m}}}
9569:{\displaystyle Y\mapsto Y^{\#}}
9502:{\displaystyle {\mathfrak {m}}}
9204:
9141:
6531:{\displaystyle {\mathfrak {g}}}
6514:is simple. The real subalgebra
6441:{\displaystyle {\mathfrak {g}}}
6386:{\displaystyle {\mathfrak {g}}}
6369:of a (real) simple Lie algebra
6358:{\displaystyle {\mathfrak {g}}}
6311:{\displaystyle {\mathfrak {g}}}
6287:{\displaystyle {\mathfrak {g}}}
6262:{\displaystyle {\mathfrak {g}}}
6219:{\displaystyle {\mathfrak {h}}}
6195:{\displaystyle {\mathfrak {h}}}
6167:{\displaystyle {\mathfrak {m}}}
3882:Space of complex structures on
2267:
2179:is compact, and by acting with
1855:{\displaystyle {\mathfrak {m}}}
1831:{\displaystyle {\mathfrak {h}}}
1803:{\displaystyle {\mathfrak {g}}}
1776:{\displaystyle {\mathfrak {h}}}
1752:{\displaystyle {\mathfrak {m}}}
1724:{\displaystyle {\mathfrak {g}}}
1700:{\displaystyle {\mathfrak {h}}}
1676:{\displaystyle {\mathfrak {h}}}
1652:{\displaystyle {\mathfrak {m}}}
1628:{\displaystyle {\mathfrak {g}}}
1604:{\displaystyle {\mathfrak {h}}}
1393:{\displaystyle {\mathfrak {g}}}
1365:{\displaystyle {\mathfrak {m}}}
1333:{\displaystyle {\mathfrak {h}}}
1297:{\displaystyle {\mathfrak {g}}}
10650:
10450:Applications and special cases
10219:
10185:
9986:
9960:
9720:
9708:
9626:
9614:
9553:
9523:
9389:
9365:
9353:
9341:
9201:
9189:
5955:
5938:
5929:double Lagrangian Grassmannian
5915:A more modern classification (
5833:
5827:
5721:
5715:
5573:
5567:
5553:
5547:
5474:
5458:
5428:
5412:
5383:
5377:
5291:
5275:
5219:
5213:
5134:
5118:
5088:
5072:
5043:
5037:
4921:
4905:
4874:
4868:
4854:
4848:
4759:
4753:
4648:
4632:
4531:
4515:
4483:
4477:
4463:
4457:
4384:
4368:
4338:
4322:
4293:
4287:
4273:
4267:
4178:
4172:
4065:
4053:
4001:
3995:
3981:
3975:
3945:
3933:
3846:
3834:
3809:
3803:
3776:
3770:
3702:
3688:
3667:
3655:
3630:
3624:
3597:
3588:
3510:
3498:
3449:
3443:
3429:
3423:
3393:
3381:
3303:
3291:
3239:
3236:
3230:
3219:
3213:
3205:
3178:
3166:
3070:
3055:
3052:
3040:
3018:
3012:
2982:
2973:
2872:
2860:
2857:
2845:
2823:
2817:
2787:
2781:
2574:is a (real) simple Lie group;
2352:is an isometry with (clearly)
2312:
2288:
2279:
2258:
2155:with a compact isotropy group
2117:
2103:
2012:
2000:
1558:
1538:
1521:
1501:
1484:
1464:
1233:
1227:
1097:
1085:
908:
899:
890:
887:
881:
875:
846:
840:
703:, there exists an isometry of
515:Galilean group representations
510:Poincaré group representations
1:
10643:
10455:Symmetric spaces and holonomy
9045:
8918:. (Selberg's assumption that
6150:is said to be irreducible if
5977:for normed division algebras
2500:is connected by assumption.)
2187:-invariant Riemannian metric
2065:such that the isotropy group
994:, and furthermore that every
505:Lorentz group representations
472:Theorem of the highest weight
11008:Spaces of constant curvature
10502:is Hermitian if and only if
9674:is the Riemannian metric on
8934:, so that in particular the
8862:, or in current terminology
8806: / SO(12,4) or E
6480:a maximal compact subgroup.
5898:{\displaystyle \mathbb {H} }
5880:which are isomorphic to the
5873:{\displaystyle \mathbb {O} }
5855:Space of subalgebras of the
2922:Space of real structures on
2886:{\displaystyle (n-1)(n+2)/2}
960:locally Riemannian symmetric
858:{\displaystyle \gamma (0)=p}
7:
11010:(5th ed.), McGraw–Hill
10866:Helgason, Sigurdur (1984),
10847:Helgason, Sigurdur (1978),
10658:(See section 5.3, page 256)
10616:
9074:. This is done by defining
9062:on the Riemannian manifold
8749: / SO(8,8) or E
6003:Lorentzian symmetric spaces
3076:{\displaystyle (n-1)(2n+1)}
2838:
2089:{\displaystyle G^{\sigma }}
1814:, equal to the identity on
1733:reductive homogeneous space
1036:
1009:is complete and Riemannian
747:pseudo-Riemannian manifolds
663:; or algebraically through
10:
11074:
10594:
10525:
10480:
10477:Hermitian symmetric spaces
10458:
8847:
6326:might not be irreducible.
6176:irreducible representation
5267:Rosenfeld projective plane
4897:Rosenfeld projective plane
4087:-dimensional subspaces of
3532:-dimensional subspaces of
3325:-dimensional subspaces of
2438:
2369:and (by differentiating) d
2343:. Then one can check that
1838:and minus the identity on
964:(globally) symmetric space
641:pseudo-Riemannian manifold
457:Lie algebra representation
20:
10909:10.1007/s00208-010-0549-8
10579: = 4, CII with
10561:quaternionic vector space
10489:Hermitian symmetric space
10483:Hermitian symmetric space
10439:the Killing form being a
9070:by combining it with the
9054:Lifting the metric tensor
6693:
4071:{\displaystyle \min(p,q)}
3516:{\displaystyle \min(p,q)}
3309:{\displaystyle \min(p,q)}
2759:Geometric interpretation
2441:List of simple Lie groups
1878:of the isometry group of
1683:-invariant complement to
11015:Wolf, Joseph A. (2007),
11006:Wolf, Joseph A. (1999),
10603:Bott periodicity theorem
10597:Bott periodicity theorem
10591:Bott periodicity theorem
8950:) is multiplicity free.
8922:should be an element of
6058:-invariant torsion-free
5999:pseudo-Riemannian metric
5989:General symmetric spaces
5921:Freudenthal magic square
990:its curvature tensor is
452:Lie group representation
10995:J. Indian Math. Society
10260:{\displaystyle i\neq j}
9036:semisimple Lie algebras
9031:is a symmetric space.
8969:, there is an isometry
8615: / SU(6,2)
8537: / SU(6,2)
8531: / SU(4,4)
8447: / SU(4,4)
8305: / Sp(3,1)
8240: / Sp(2,2)
8164: / Sp(3,1)
8090: / Sp(2,2)
7960: / SO(8,1)
7923: / SO(5,4)
5925:Lagrangian Grassmannian
5804:{\displaystyle G_{2}\,}
5744:Cayley projective plane
5686:{\displaystyle F_{4}\,}
5524:{\displaystyle F_{4}\,}
5341:{\displaystyle E_{8}\,}
5184:{\displaystyle E_{8}\,}
5001:{\displaystyle E_{7}\,}
4825:{\displaystyle E_{7}\,}
4730:{\displaystyle E_{7}\,}
4610:{\displaystyle F_{4}\,}
4581:{\displaystyle E_{6}\,}
4507:Cayley projective plane
4434:{\displaystyle E_{6}\,}
4244:{\displaystyle E_{6}\,}
4149:{\displaystyle E_{6}\,}
1886:acting transitively on
1611:is a Lie subalgebra of
1141:is a homogeneous space
477:Borel–Weil–Bott theorem
10736:Besse, Arthur Lancelot
10422:
10285:
10261:
10232:
10166:
10099:
10072:
9859:
9801:
9777:
9727:
9692:
9668:
9633:
9570:
9537:
9503:
9471:
9430:
9396:
9325:
9324:{\displaystyle T_{p}M}
9295:
9253:
8936:unitary representation
8864:weakly symmetric space
8850:Weakly symmetric space
8742: / SO(16)
8716: / SO(16)
6664:
6606:
6571:
6532:
6508:
6442:
6418:
6387:
6359:
6312:
6288:
6263:
6220:
6196:
6168:
6141:
6087:Classification results
5971:
5917:Huang & Leung 2010
5899:
5874:
5841:
5805:
5771:
5729:
5687:
5653:
5621:
5581:
5525:
5491:
5445:
5391:
5342:
5308:
5252:
5185:
5151:
5105:
5051:
5002:
4968:
4938:
4882:
4826:
4784:
4731:
4697:
4665:
4611:
4582:
4548:
4491:
4435:
4401:
4355:
4301:
4245:
4203:
4150:
4116:
4072:
4036:
4009:
3953:
3905:
3874:
3853:
3852:{\displaystyle n(n+1)}
3817:
3784:
3743:
3709:
3674:
3673:{\displaystyle n(n-1)}
3638:
3605:
3561:
3517:
3481:
3457:
3401:
3354:
3310:
3274:
3247:
3186:
3138:
3104:
3077:
3026:
2990:
2945:
2914:
2887:
2831:
2795:
2640:exceptional Lie groups
2476:, first note that the
2322:
2171:at the identity coset
2134:
2090:
2048:
1856:
1832:
1804:
1777:
1753:
1725:
1701:
1677:
1653:
1629:
1605:
1578:
1445:
1394:
1376:is an automorphism of
1366:
1334:
1298:
1272:As an automorphism of
1252:
1168:is an automorphism of
1104:
1103:{\displaystyle L(2,1)}
918:
859:
822:if it fixes the point
735:
734:{\displaystyle T_{p}M}
639:(or more generally, a
375:Semisimple Lie algebra
330:Adjoint representation
11043:Differential geometry
10946:Loos, Ottmar (1969),
10937:Loos, Ottmar (1969),
10897:Mathematische Annalen
10423:
10286:
10262:
10233:
10167:
10100:
10073:
9860:
9802:
9778:
9735:generalized transpose
9728:
9693:
9669:
9634:
9571:
9538:
9504:
9472:
9431:
9397:
9326:
9296:
9254:
8894:there is an isometry
8679: / SO(16,
8440: / SU(8)
8407: / SU(8)
8366: / SO(12,
8083: / Sp(4)
8044: / Sp(4)
8008: / SO(10,
7942: / SO(9)
7872: / SO(9)
6665:
6607:
6572:
6533:
6509:
6443:
6419:
6388:
6360:
6313:
6289:
6264:
6221:
6197:
6169:
6142:
5972:
5900:
5875:
5842:
5806:
5772:
5730:
5688:
5654:
5622:
5582:
5526:
5492:
5446:
5392:
5343:
5309:
5253:
5186:
5152:
5106:
5052:
5003:
4969:
4939:
4883:
4827:
4785:
4732:
4698:
4666:
4612:
4583:
4549:
4492:
4436:
4402:
4356:
4302:
4246:
4204:
4151:
4117:
4073:
4037:
4010:
3954:
3906:
3875:
3854:
3818:
3785:
3744:
3710:
3675:
3639:
3606:
3562:
3518:
3482:
3458:
3402:
3355:
3311:
3275:
3248:
3187:
3139:
3105:
3078:
3027:
2991:
2946:
2915:
2888:
2832:
2796:
2714:Classification result
2504:Classification scheme
2335:is the involution of
2323:
2147:is a symmetric space
2135:
2091:
2049:
1951:at any point) and so
1857:
1833:
1805:
1778:
1754:
1726:
1702:
1678:
1654:
1630:
1606:
1579:
1446:
1395:
1367:
1335:
1299:
1253:
1149:where the stabilizer
1105:
1017:acts transitively on
919:
860:
814:of a neighborhood of
736:
677:representation theory
673:differential geometry
444:Representation theory
10628:Relative root system
10298:
10271:
10245:
10182:
10112:
10085:
9875:
9814:
9787:
9741:
9702:
9678:
9646:
9583:
9547:
9513:
9489:
9440:
9410:
9335:
9305:
9266:
9081:
8810: / Sk(8,
8753: / Sk(8,
8632: / Sk(6,
8605: / SL(4,
8550: / Sk(6,
8484: / Sk(6,
8463: / SL(4,
8453: / SL(8,
8355: / SL(8,
8312: / SL(3,
8204: / Sk(5,
8170: / Sp(8,
8121: / SL(3,
8107: / SL(6,
8096: / Sp(8,
7993: / SL(6,
7982: / Sp(8,
7900: / Sp(6,
7837: / SO(9,
7820: / Sp(6,
7788: / SL(2,
7730: / SL(2,
7065: / S(GL(
7017:) / Sp(2
6816: / S(GL(
6735: / S(GL(
6643:
6585:
6550:
6518:
6487:
6428:
6424:is not simple, then
6397:
6373:
6345:
6298:
6274:
6249:
6206:
6182:
6154:
6107:
6027:anti-de Sitter space
5935:
5887:
5862:
5816:
5787:
5749:
5698:
5669:
5631:
5599:
5536:
5507:
5455:
5409:
5353:
5324:
5272:
5196:
5167:
5115:
5069:
5013:
4984:
4948:
4902:
4837:
4808:
4742:
4713:
4675:
4629:
4593:
4564:
4512:
4446:
4417:
4365:
4319:
4256:
4227:
4161:
4132:
4091:
4047:
4020:
3964:
3922:
3886:
3864:
3828:
3795:
3759:
3721:
3685:
3649:
3616:
3577:
3536:
3492:
3468:
3412:
3370:
3329:
3285:
3258:
3197:
3155:
3116:
3088:
3037:
3001:
2962:
2926:
2898:
2842:
2806:
2770:
2683:is such a group and
2588:. For compact type,
2492:is connected by the
2239:
2100:
2073:
1988:
1842:
1818:
1790:
1763:
1739:
1711:
1687:
1663:
1639:
1615:
1591:
1461:
1411:
1380:
1352:
1320:
1284:
1193:
1121:Algebraic definition
1115:anti-de Sitter space
1079:
992:covariantly constant
869:
834:
794:Geometric definition
715:
21:For other uses, see
11048:Riemannian geometry
10746:, Springer-Verlag,
10728:10.24033/asens.1054
10009:
9895:
9831:
9477:positive-definite.
8961:and tangent vector
7621: / Sp(2
7591:) / GL(
7559:) / Sp(
7441: / Sp(2
7315:) / GL(
7275:) / SO(
6977:) / GL(
6933: / S(U(
6919:) / Sk(
6483:Thus we may assume
4035:{\displaystyle 4pq}
3273:{\displaystyle 2pq}
3103:{\displaystyle n-1}
2913:{\displaystyle n-1}
2547:sectional curvature
830:is a geodesic with
810:. A diffeomorphism
657:Riemannian geometry
637:Riemannian manifold
589:Table of Lie groups
430:Compact Lie algebra
11058:Homogeneous spaces
10874:, Academic Press,
10851:, Academic Press,
10822:10.24033/bsmf.1113
10800:10.24033/bsmf.1105
10742:Einstein Manifolds
10686:10.1007/BF01236659
10583: = 1 or
10575: = 4 or
10567: = 2 or
10418:
10281:
10267:. For the case of
10257:
10231:{\displaystyle =0}
10228:
10162:
10095:
10068:
9995:
9881:
9855:
9817:
9797:
9773:
9723:
9688:
9664:
9629:
9566:
9533:
9499:
9485:The tangent space
9467:
9426:
9392:
9321:
9291:
9249:
9244:
9240:
9023:is independent of
8999:the derivative of
7682: / GL(
7643: / Sp(
7515: / Sp(
7474: / GL(
7407: / SL(
7371: / SO(
7349: / Sk(
7229: / SO(
7195: / GL(
7162: / SO(
7109: / Sp(
7095: / GL(
7051: / Sk(
6991: / Sp(
6901: / SO(
6856: / Sp(
6838: / GL(
6802: / SO(
6768: / Sp(
6721: / SO(
6660:
6602:
6567:
6528:
6504:
6460:is a real form of
6438:
6414:
6383:
6355:
6308:
6284:
6259:
6216:
6192:
6164:
6137:
6015:general relativity
5967:
5895:
5882:quaternion algebra
5870:
5837:
5801:
5767:
5725:
5683:
5649:
5617:
5577:
5521:
5487:
5441:
5387:
5338:
5304:
5248:
5181:
5147:
5101:
5047:
4998:
4964:
4934:
4878:
4822:
4780:
4727:
4693:
4661:
4607:
4578:
4544:
4487:
4431:
4397:
4351:
4297:
4241:
4199:
4146:
4112:
4068:
4032:
4005:
3949:
3901:
3870:
3849:
3813:
3780:
3739:
3705:
3670:
3634:
3601:
3557:
3513:
3480:{\displaystyle pq}
3477:
3453:
3397:
3350:
3306:
3270:
3243:
3182:
3134:
3100:
3073:
3022:
2986:
2941:
2910:
2883:
2827:
2791:
2513:symmetric spaces.
2318:
2130:
2086:
2044:
1852:
1828:
1800:
1773:
1749:
1721:
1697:
1673:
1649:
1625:
1601:
1574:
1441:
1390:
1362:
1330:
1308:, also denoted by
1294:
1248:
1100:
914:
855:
767:by a Lie subgroup
731:
653:inversion symmetry
361:Affine Lie algebra
351:Simple Lie algebra
92:Special orthogonal
11026:978-0-8218-4289-8
10840:978-0-8218-0711-8
10703:978-0-8218-0609-8
10668:Akhiezer, D. N.;
10638:Cartan involution
10388:
10334:
10105:into eigenspaces
9422:
9239:
8841:
8840:
8661:
8660:
8337:
8336:
7964:
7963:
7800:
7799:
7694:
7693:
7657: / U(
7607: = Sp(2
7573: / U(
7427: = Sp(2
7419:
7418:
7385: / U(
7289: / U(
7140:
7139:
6688:Cartan involution
6100:with Lie algebra
6060:affine connection
5908:
5907:
3873:{\displaystyle n}
3528:of oriented real
2586:simple Lie groups
1918:of the action of
1055:hyperbolic spaces
1051:projective spaces
820:geodesic symmetry
681:harmonic analysis
625:
624:
425:Split Lie algebra
388:Cartan subalgebra
250:
249:
141:Simple Lie groups
11065:
11029:
11011:
11002:
10986:
10951:
10942:
10932:
10912:
10894:
10884:
10873:
10861:
10843:
10825:
10824:
10803:
10802:
10778:
10756:
10745:
10731:
10730:
10706:
10688:
10660:
10654:
10611:orthogonal group
10557:unit quaternions
10517:
10441:Casimir operator
10427:
10425:
10424:
10419:
10417:
10416:
10415:
10414:
10409:
10408:
10400:
10389:
10387:
10386:
10374:
10363:
10362:
10361:
10360:
10355:
10354:
10346:
10335:
10333:
10332:
10320:
10290:
10288:
10287:
10282:
10280:
10279:
10266:
10264:
10263:
10258:
10237:
10235:
10234:
10229:
10218:
10217:
10212:
10211:
10201:
10200:
10195:
10194:
10171:
10169:
10168:
10163:
10161:
10160:
10155:
10154:
10138:
10137:
10132:
10131:
10121:
10120:
10104:
10102:
10101:
10096:
10094:
10093:
10077:
10075:
10074:
10069:
10064:
10063:
10051:
10050:
10038:
10037:
10022:
10021:
10008:
10003:
9985:
9984:
9972:
9971:
9950:
9949:
9937:
9936:
9924:
9923:
9908:
9907:
9894:
9889:
9864:
9862:
9861:
9856:
9854:
9853:
9844:
9843:
9830:
9825:
9806:
9804:
9803:
9798:
9796:
9795:
9782:
9780:
9779:
9774:
9772:
9771:
9753:
9752:
9732:
9730:
9729:
9724:
9697:
9695:
9694:
9689:
9687:
9686:
9673:
9671:
9670:
9665:
9638:
9636:
9635:
9630:
9604:
9603:
9575:
9573:
9572:
9567:
9565:
9564:
9542:
9540:
9539:
9534:
9532:
9531:
9522:
9521:
9508:
9506:
9505:
9500:
9498:
9497:
9476:
9474:
9473:
9468:
9466:
9465:
9464:
9435:
9433:
9432:
9427:
9420:
9419:
9418:
9401:
9399:
9398:
9393:
9330:
9328:
9327:
9322:
9317:
9316:
9300:
9298:
9297:
9292:
9290:
9289:
9258:
9256:
9255:
9250:
9248:
9247:
9241:
9237:
9225:
9224:
9178:
9177:
9165:
9164:
9140:
9139:
9107:
9106:
9105:
8882:such that given
8874:and an isometry
8829: / E
8825:×SU(2) or E
8821: / E
8795: / E
8776: / E
8764: / E
8723: / E
8690: / E
8663:
8662:
8653: / E
8643: / E
8594: / E
8575: / E
8565: / E
8505: / E
8495: / E
8421: / E
8381: / E
8339:
8338:
8330: / F
8295: / F
8277: / F
8215: / F
8139: / F
8065: / F
8023: / F
7966:
7965:
7802:
7801:
7710:
7709:
7706:
7490: = Sp(
7421:
7420:
7331: = Sk(
7142:
7141:
6701:
6700:
6669:
6667:
6666:
6661:
6659:
6658:
6653:
6652:
6611:
6609:
6608:
6603:
6601:
6600:
6595:
6594:
6576:
6574:
6573:
6568:
6566:
6565:
6560:
6559:
6537:
6535:
6534:
6529:
6527:
6526:
6513:
6511:
6510:
6505:
6503:
6502:
6497:
6496:
6447:
6445:
6444:
6439:
6437:
6436:
6423:
6421:
6420:
6415:
6413:
6412:
6407:
6406:
6392:
6390:
6389:
6384:
6382:
6381:
6364:
6362:
6361:
6356:
6354:
6353:
6317:
6315:
6314:
6309:
6307:
6306:
6293:
6291:
6290:
6285:
6283:
6282:
6268:
6266:
6265:
6260:
6258:
6257:
6225:
6223:
6222:
6217:
6215:
6214:
6201:
6199:
6198:
6193:
6191:
6190:
6173:
6171:
6170:
6165:
6163:
6162:
6146:
6144:
6143:
6138:
6136:
6135:
6126:
6125:
6116:
6115:
6053:
6037: + 1.
5976:
5974:
5973:
5968:
5963:
5962:
5953:
5945:
5931:of subspaces of
5911:As Grassmannians
5904:
5902:
5901:
5896:
5894:
5879:
5877:
5876:
5871:
5869:
5857:octonion algebra
5846:
5844:
5843:
5838:
5826:
5810:
5808:
5807:
5802:
5799:
5798:
5776:
5774:
5773:
5768:
5766:
5765:
5756:
5734:
5732:
5731:
5726:
5714:
5692:
5690:
5689:
5684:
5681:
5680:
5658:
5656:
5655:
5650:
5648:
5647:
5638:
5626:
5624:
5623:
5618:
5616:
5615:
5606:
5586:
5584:
5583:
5578:
5566:
5546:
5530:
5528:
5527:
5522:
5519:
5518:
5496:
5494:
5493:
5488:
5486:
5485:
5473:
5465:
5450:
5448:
5447:
5442:
5440:
5439:
5427:
5419:
5396:
5394:
5393:
5388:
5376:
5365:
5364:
5347:
5345:
5344:
5339:
5336:
5335:
5313:
5311:
5310:
5305:
5303:
5302:
5290:
5282:
5257:
5255:
5254:
5249:
5243:
5226:
5212:
5190:
5188:
5187:
5182:
5179:
5178:
5156:
5154:
5153:
5148:
5146:
5145:
5133:
5125:
5110:
5108:
5107:
5102:
5100:
5099:
5087:
5079:
5056:
5054:
5053:
5048:
5036:
5025:
5024:
5007:
5005:
5004:
4999:
4996:
4995:
4973:
4971:
4970:
4965:
4963:
4955:
4943:
4941:
4940:
4935:
4933:
4932:
4920:
4912:
4887:
4885:
4884:
4879:
4867:
4847:
4831:
4829:
4828:
4823:
4820:
4819:
4789:
4787:
4786:
4781:
4766:
4752:
4736:
4734:
4733:
4728:
4725:
4724:
4702:
4700:
4699:
4694:
4692:
4691:
4686:
4670:
4668:
4667:
4662:
4660:
4659:
4647:
4639:
4616:
4614:
4613:
4608:
4605:
4604:
4587:
4585:
4584:
4579:
4576:
4575:
4553:
4551:
4550:
4545:
4543:
4542:
4530:
4522:
4496:
4494:
4493:
4488:
4476:
4456:
4440:
4438:
4437:
4432:
4429:
4428:
4406:
4404:
4403:
4398:
4396:
4395:
4383:
4375:
4360:
4358:
4357:
4352:
4350:
4349:
4337:
4329:
4306:
4304:
4303:
4298:
4286:
4266:
4250:
4248:
4247:
4242:
4239:
4238:
4208:
4206:
4205:
4200:
4185:
4171:
4155:
4153:
4152:
4147:
4144:
4143:
4121:
4119:
4118:
4113:
4111:
4110:
4099:
4083:of quaternionic
4077:
4075:
4074:
4069:
4041:
4039:
4038:
4033:
4014:
4012:
4011:
4006:
3994:
3974:
3958:
3956:
3955:
3950:
3932:
3910:
3908:
3907:
3902:
3900:
3899:
3894:
3879:
3877:
3876:
3871:
3858:
3856:
3855:
3850:
3822:
3820:
3819:
3814:
3802:
3789:
3787:
3786:
3781:
3769:
3748:
3746:
3745:
3740:
3738:
3737:
3729:
3714:
3712:
3711:
3708:{\displaystyle }
3706:
3698:
3679:
3677:
3676:
3671:
3643:
3641:
3640:
3635:
3623:
3610:
3608:
3607:
3602:
3587:
3566:
3564:
3563:
3558:
3556:
3555:
3544:
3522:
3520:
3519:
3514:
3486:
3484:
3483:
3478:
3462:
3460:
3459:
3454:
3442:
3422:
3406:
3404:
3403:
3398:
3380:
3359:
3357:
3356:
3351:
3349:
3348:
3337:
3315:
3313:
3312:
3307:
3279:
3277:
3276:
3271:
3252:
3250:
3249:
3244:
3229:
3212:
3204:
3191:
3189:
3188:
3183:
3165:
3143:
3141:
3140:
3135:
3133:
3132:
3124:
3109:
3107:
3106:
3101:
3082:
3080:
3079:
3074:
3031:
3029:
3028:
3023:
3011:
2995:
2993:
2992:
2987:
2972:
2950:
2948:
2947:
2942:
2940:
2939:
2934:
2919:
2917:
2916:
2911:
2892:
2890:
2889:
2884:
2879:
2836:
2834:
2833:
2828:
2816:
2800:
2798:
2797:
2792:
2780:
2737:
2736:
2553:Non-compact type
2327:
2325:
2324:
2319:
2311:
2303:
2302:
2275:
2251:
2250:
2219:
2139:
2137:
2136:
2131:
2125:
2124:
2115:
2114:
2095:
2093:
2092:
2087:
2085:
2084:
2053:
2051:
2050:
2045:
2043:
2042:
2024:
2023:
1861:
1859:
1858:
1853:
1851:
1850:
1837:
1835:
1834:
1829:
1827:
1826:
1809:
1807:
1806:
1801:
1799:
1798:
1782:
1780:
1779:
1774:
1772:
1771:
1758:
1756:
1755:
1750:
1748:
1747:
1730:
1728:
1727:
1722:
1720:
1719:
1706:
1704:
1703:
1698:
1696:
1695:
1682:
1680:
1679:
1674:
1672:
1671:
1658:
1656:
1655:
1650:
1648:
1647:
1634:
1632:
1631:
1626:
1624:
1623:
1610:
1608:
1607:
1602:
1600:
1599:
1583:
1581:
1580:
1575:
1570:
1569:
1557:
1556:
1547:
1546:
1533:
1532:
1520:
1519:
1510:
1509:
1496:
1495:
1483:
1482:
1473:
1472:
1450:
1448:
1447:
1442:
1440:
1439:
1430:
1429:
1420:
1419:
1399:
1397:
1396:
1391:
1389:
1388:
1371:
1369:
1368:
1363:
1361:
1360:
1339:
1337:
1336:
1331:
1329:
1328:
1303:
1301:
1300:
1295:
1293:
1292:
1257:
1255:
1254:
1249:
1205:
1204:
1109:
1107:
1106:
1101:
996:simply connected
974:Basic properties
923:
921:
920:
915:
864:
862:
861:
856:
818:is said to be a
740:
738:
737:
732:
727:
726:
667:, which allowed
617:
610:
603:
560:Claude Chevalley
417:Complexification
260:Other Lie groups
146:
145:
54:Classical groups
46:
28:
27:
11073:
11072:
11068:
11067:
11066:
11064:
11063:
11062:
11033:
11032:
11027:
10976:10.2307/2372398
10930:
10892:
10882:
10859:
10841:
10776:
10754:
10704:
10664:
10663:
10655:
10651:
10646:
10619:
10599:
10593:
10547: /
10530:
10524:
10512:
10498: /
10485:
10479:
10463:
10457:
10452:
10410:
10404:
10403:
10402:
10401:
10391:
10390:
10382:
10378:
10373:
10356:
10350:
10349:
10348:
10347:
10337:
10336:
10328:
10324:
10319:
10299:
10296:
10295:
10275:
10274:
10272:
10269:
10268:
10246:
10243:
10242:
10213:
10207:
10206:
10205:
10196:
10190:
10189:
10188:
10183:
10180:
10179:
10156:
10150:
10149:
10148:
10133:
10127:
10126:
10125:
10116:
10115:
10113:
10110:
10109:
10089:
10088:
10086:
10083:
10082:
10059:
10055:
10046:
10042:
10033:
10029:
10017:
10013:
10004:
9999:
9980:
9976:
9967:
9963:
9945:
9941:
9932:
9928:
9919:
9915:
9903:
9899:
9890:
9885:
9876:
9873:
9872:
9849:
9845:
9839:
9835:
9826:
9821:
9815:
9812:
9811:
9791:
9790:
9788:
9785:
9784:
9767:
9763:
9748:
9744:
9742:
9739:
9738:
9703:
9700:
9699:
9682:
9681:
9679:
9676:
9675:
9647:
9644:
9643:
9599:
9595:
9584:
9581:
9580:
9560:
9556:
9548:
9545:
9544:
9527:
9526:
9517:
9516:
9514:
9511:
9510:
9493:
9492:
9490:
9487:
9486:
9483:
9460:
9459:
9455:
9441:
9438:
9437:
9414:
9413:
9411:
9408:
9407:
9336:
9333:
9332:
9312:
9308:
9306:
9303:
9302:
9285:
9281:
9267:
9264:
9263:
9243:
9242:
9235:
9233:
9227:
9226:
9220:
9219:
9205:
9180:
9179:
9173:
9172:
9160:
9156:
9142:
9135:
9131:
9112:
9111:
9101:
9100:
9096:
9082:
9079:
9078:
9056:
9048:
8977:, depending on
8852:
8846:
8832:
8828:
8824:
8820:
8809:
8805:
8798:
8794:
8788:
8779:
8775:
8767:
8763:
8752:
8748:
8741:
8735:
8726:
8722:
8715:
8706:
8693:
8689:
8678:
8669:
8656:
8652:
8648:
8646:
8642:
8631:
8627:
8621:
8614:
8610:
8604:
8597:
8593:
8587:
8578:
8574:
8570:
8568:
8564:
8549:
8545:
8543:
8536:
8532:
8530:
8523:
8517:
8508:
8504:
8500:
8498:
8494:
8483:
8479:
8473:
8462:
8458:
8452:
8448:
8446:
8439:
8433:
8424:
8420:
8413:
8406:
8397:
8384:
8380:
8365:
8354:
8345:
8333:
8329:
8322:
8311:
8304:
8298:
8294:
8288:
8280:
8276:
8265:
8263:
8252:
8248:
8246:
8239:
8232:
8226:
8218:
8214:
8203:
8199:
8197:
8186:
8182:
8180:
8169:
8165:
8163:
8156:
8150:
8142:
8138:
8131:
8120:
8116:
8106:
8095:
8091:
8089:
8082:
8076:
8068:
8064:
8057:
8050:
8043:
8034:
8026:
8022:
8007:
7992:
7981:
7972:
7959:
7950:
7941:
7933:
7922:
7913:
7909:
7899:
7890:
7882:
7871:
7862:
7851:
7836:
7819:
7808:
7787:
7778:
7770:
7759:
7748:
7729:
7718:
7701:
7677:
7638:
7582:
7550:
7548:
7539:
7530:
7521:
7402:
7366:
7306:
7266:
7264:
7255:
7246:
7237:
7090:
7008:
6968:
6966:
6957:
6948:
6939:
6910:
6833:
6696:
6654:
6648:
6647:
6646:
6644:
6641:
6640:
6612:commuting with
6596:
6590:
6589:
6588:
6586:
6583:
6582:
6561:
6555:
6554:
6553:
6551:
6548:
6547:
6522:
6521:
6519:
6516:
6515:
6498:
6492:
6491:
6490:
6488:
6485:
6484:
6468: /
6452: /
6432:
6431:
6429:
6426:
6425:
6408:
6402:
6401:
6400:
6398:
6395:
6394:
6377:
6376:
6374:
6371:
6370:
6349:
6348:
6346:
6343:
6342:
6322: /
6302:
6301:
6299:
6296:
6295:
6278:
6277:
6275:
6272:
6271:
6253:
6252:
6250:
6247:
6246:
6241: /
6210:
6209:
6207:
6204:
6203:
6186:
6185:
6183:
6180:
6179:
6158:
6157:
6155:
6152:
6151:
6131:
6130:
6121:
6120:
6111:
6110:
6108:
6105:
6104:
6096: /
6089:
6080:universal cover
6049: /
6041:
6023:De Sitter space
6019:Minkowski space
5991:
5958:
5954:
5949:
5941:
5936:
5933:
5932:
5913:
5890:
5888:
5885:
5884:
5865:
5863:
5860:
5859:
5819:
5817:
5814:
5813:
5794:
5790:
5788:
5785:
5784:
5761:
5757:
5752:
5750:
5747:
5746:
5701:
5699:
5696:
5695:
5676:
5672:
5670:
5667:
5666:
5643:
5639:
5634:
5632:
5629:
5628:
5611:
5607:
5602:
5600:
5597:
5596:
5559:
5539:
5537:
5534:
5533:
5514:
5510:
5508:
5505:
5504:
5481:
5477:
5469:
5461:
5456:
5453:
5452:
5435:
5431:
5423:
5415:
5410:
5407:
5406:
5369:
5360:
5356:
5354:
5351:
5350:
5331:
5327:
5325:
5322:
5321:
5298:
5294:
5286:
5278:
5273:
5270:
5269:
5233:
5222:
5199:
5197:
5194:
5193:
5174:
5170:
5168:
5165:
5164:
5141:
5137:
5129:
5121:
5116:
5113:
5112:
5095:
5091:
5083:
5075:
5070:
5067:
5066:
5029:
5020:
5016:
5014:
5011:
5010:
4991:
4987:
4985:
4982:
4981:
4959:
4951:
4949:
4946:
4945:
4928:
4924:
4916:
4908:
4903:
4900:
4899:
4860:
4840:
4838:
4835:
4834:
4815:
4811:
4809:
4806:
4805:
4762:
4745:
4743:
4740:
4739:
4720:
4716:
4714:
4711:
4710:
4687:
4679:
4678:
4676:
4673:
4672:
4655:
4651:
4643:
4635:
4630:
4627:
4626:
4600:
4596:
4594:
4591:
4590:
4571:
4567:
4565:
4562:
4561:
4538:
4534:
4526:
4518:
4513:
4510:
4509:
4469:
4449:
4447:
4444:
4443:
4424:
4420:
4418:
4415:
4414:
4391:
4387:
4379:
4371:
4366:
4363:
4362:
4345:
4341:
4333:
4325:
4320:
4317:
4316:
4279:
4259:
4257:
4254:
4253:
4234:
4230:
4228:
4225:
4224:
4181:
4164:
4162:
4159:
4158:
4139:
4135:
4133:
4130:
4129:
4100:
4095:
4094:
4092:
4089:
4088:
4048:
4045:
4044:
4021:
4018:
4017:
3987:
3967:
3965:
3962:
3961:
3925:
3923:
3920:
3919:
3895:
3890:
3889:
3887:
3884:
3883:
3865:
3862:
3861:
3829:
3826:
3825:
3798:
3796:
3793:
3792:
3762:
3760:
3757:
3756:
3730:
3725:
3724:
3722:
3719:
3718:
3694:
3686:
3683:
3682:
3650:
3647:
3646:
3619:
3617:
3614:
3613:
3580:
3578:
3575:
3574:
3545:
3540:
3539:
3537:
3534:
3533:
3493:
3490:
3489:
3469:
3466:
3465:
3435:
3415:
3413:
3410:
3409:
3373:
3371:
3368:
3367:
3338:
3333:
3332:
3330:
3327:
3326:
3286:
3283:
3282:
3259:
3256:
3255:
3225:
3208:
3200:
3198:
3195:
3194:
3158:
3156:
3153:
3152:
3125:
3120:
3119:
3117:
3114:
3113:
3089:
3086:
3085:
3038:
3035:
3034:
3004:
3002:
2999:
2998:
2965:
2963:
2960:
2959:
2935:
2930:
2929:
2927:
2924:
2923:
2899:
2896:
2895:
2875:
2843:
2840:
2839:
2809:
2807:
2804:
2803:
2773:
2771:
2768:
2767:
2722: /
2716:
2675:
2668:
2661:
2654:
2647:
2638:) and the five
2533:Euclidean space
2506:
2478:universal cover
2443:
2437:
2400:was arbitrary,
2395:
2383:
2377:
2360:
2351:
2304:
2295:
2291:
2268:
2246:
2242:
2240:
2237:
2236:
2211:
2206: /
2195: /
2167: /
2151: /
2120:
2116:
2110:
2106:
2101:
2098:
2097:
2080:
2076:
2074:
2071:
2070:
2038:
2034:
2019:
2015:
1989:
1986:
1985:
1972:
1960:
1943:
1868:
1846:
1845:
1843:
1840:
1839:
1822:
1821:
1819:
1816:
1815:
1794:
1793:
1791:
1788:
1787:
1767:
1766:
1764:
1761:
1760:
1743:
1742:
1740:
1737:
1736:
1715:
1714:
1712:
1709:
1708:
1691:
1690:
1688:
1685:
1684:
1667:
1666:
1664:
1661:
1660:
1643:
1642:
1640:
1637:
1636:
1619:
1618:
1616:
1613:
1612:
1595:
1594:
1592:
1589:
1588:
1565:
1564:
1552:
1551:
1542:
1541:
1528:
1527:
1515:
1514:
1505:
1504:
1491:
1490:
1478:
1477:
1468:
1467:
1462:
1459:
1458:
1435:
1434:
1425:
1424:
1415:
1414:
1412:
1409:
1408:
1400:, this gives a
1384:
1383:
1381:
1378:
1377:
1356:
1355:
1353:
1350:
1349:
1324:
1323:
1321:
1318:
1317:
1288:
1287:
1285:
1282:
1281:
1200:
1196:
1194:
1191:
1190:
1181:
1145: /
1135:symmetric space
1129:be a connected
1123:
1080:
1077:
1076:
1066:Riemann surface
1043:Euclidean space
1039:
1029:is connected).
976:
870:
867:
866:
835:
832:
831:
796:
760:of a connected
756: /
722:
718:
716:
713:
712:
633:symmetric space
621:
576:
575:
574:
545:Wilhelm Killing
529:
521:
520:
519:
494:
483:
482:
481:
446:
436:
435:
434:
421:
405:
383:Dynkin diagrams
377:
367:
366:
365:
347:
325:Exponential map
314:
304:
303:
302:
283:Conformal group
262:
252:
251:
243:
235:
227:
219:
211:
192:
182:
172:
162:
143:
133:
132:
131:
112:Special unitary
56:
26:
19:
12:
11:
5:
11071:
11061:
11060:
11055:
11050:
11045:
11031:
11030:
11025:
11012:
11003:
10987:
10964:Amer. J. Math.
10952:
10943:
10934:
10928:
10913:
10885:
10880:
10863:
10857:
10844:
10839:
10826:
10804:
10779:
10774:
10758:
10752:
10732:
10711:Berger, Marcel
10707:
10702:
10689:
10674:Transf. Groups
10670:Vinberg, E. B.
10662:
10661:
10648:
10647:
10645:
10642:
10641:
10640:
10635:
10633:Satake diagram
10630:
10625:
10618:
10615:
10609:of the stable
10595:Main article:
10592:
10589:
10526:Main article:
10523:
10520:
10481:Main article:
10478:
10475:
10467:holonomy group
10461:Holonomy group
10459:Main article:
10456:
10453:
10451:
10448:
10429:
10428:
10413:
10407:
10399:
10396:
10393:
10385:
10381:
10377:
10372:
10369:
10366:
10359:
10353:
10345:
10342:
10339:
10331:
10327:
10323:
10318:
10315:
10312:
10309:
10306:
10303:
10278:
10256:
10253:
10250:
10239:
10238:
10227:
10224:
10221:
10216:
10210:
10204:
10199:
10193:
10187:
10173:
10172:
10159:
10153:
10147:
10144:
10141:
10136:
10130:
10124:
10119:
10092:
10079:
10078:
10067:
10062:
10058:
10054:
10049:
10045:
10041:
10036:
10032:
10028:
10025:
10020:
10016:
10012:
10007:
10002:
9998:
9994:
9991:
9988:
9983:
9979:
9975:
9970:
9966:
9962:
9959:
9956:
9953:
9948:
9944:
9940:
9935:
9931:
9927:
9922:
9918:
9914:
9911:
9906:
9902:
9898:
9893:
9888:
9884:
9880:
9866:
9865:
9852:
9848:
9842:
9838:
9834:
9829:
9824:
9820:
9794:
9770:
9766:
9762:
9759:
9756:
9751:
9747:
9722:
9719:
9716:
9713:
9710:
9707:
9685:
9663:
9660:
9657:
9654:
9651:
9640:
9639:
9628:
9625:
9622:
9619:
9616:
9613:
9610:
9607:
9602:
9598:
9594:
9591:
9588:
9563:
9559:
9555:
9552:
9530:
9525:
9520:
9496:
9482:
9479:
9463:
9458:
9454:
9451:
9448:
9445:
9425:
9417:
9391:
9388:
9385:
9382:
9379:
9376:
9373:
9370:
9367:
9364:
9361:
9358:
9355:
9352:
9349:
9346:
9343:
9340:
9320:
9315:
9311:
9288:
9284:
9280:
9277:
9274:
9271:
9260:
9259:
9246:
9234:
9232:
9229:
9228:
9223:
9218:
9215:
9212:
9209:
9206:
9203:
9200:
9197:
9194:
9191:
9188:
9185:
9182:
9181:
9176:
9171:
9168:
9163:
9159:
9155:
9152:
9149:
9146:
9143:
9138:
9134:
9130:
9127:
9124:
9121:
9118:
9117:
9115:
9110:
9104:
9099:
9095:
9092:
9089:
9086:
9055:
9052:
9047:
9044:
9038:, is given in
9017:
9016:
8997:
8928:Ernest Vinberg
8848:Main article:
8845:
8842:
8839:
8838:
8830:
8826:
8822:
8818:
8815:
8807:
8803:
8800:
8796:
8792:
8789:
8786:
8782:
8781:
8777:
8773:
8765:
8761:
8758:
8750:
8746:
8743:
8739:
8736:
8733:
8729:
8728:
8724:
8720:
8717:
8713:
8710:
8707:
8704:
8700:
8699:
8691:
8687:
8684:
8676:
8673:
8670:
8667:
8659:
8658:
8657:×SO(1,1)
8654:
8650:
8644:
8640:
8637:
8629:
8619:
8616:
8612:
8602:
8599:
8595:
8591:
8588:
8585:
8581:
8580:
8576:
8572:
8566:
8562:
8559:
8547:
8541:
8538:
8534:
8528:
8525:
8521:
8518:
8515:
8511:
8510:
8506:
8502:
8499:×SO(1,1)
8496:
8492:
8489:
8481:
8471:
8468:
8460:
8450:
8444:
8441:
8437:
8434:
8431:
8427:
8426:
8422:
8418:
8415:
8411:
8408:
8404:
8401:
8398:
8395:
8391:
8390:
8382:
8378:
8375:
8363:
8360:
8352:
8349:
8346:
8343:
8335:
8334:
8331:
8327:
8324:
8320:
8317:
8309:
8306:
8302:
8299:
8296:
8292:
8289:
8286:
8282:
8281:
8278:
8274:
8271:
8261:
8258:
8250:
8244:
8241:
8237:
8234:
8230:
8227:
8224:
8220:
8219:
8216:
8212:
8209:
8201:
8195:
8192:
8184:
8178:
8175:
8167:
8161:
8158:
8154:
8151:
8148:
8144:
8143:
8140:
8136:
8133:
8129:
8126:
8118:
8104:
8101:
8093:
8087:
8084:
8080:
8077:
8074:
8070:
8069:
8066:
8062:
8059:
8055:
8052:
8048:
8045:
8041:
8038:
8035:
8032:
8028:
8027:
8024:
8020:
8017:
8005:
8002:
7990:
7987:
7979:
7976:
7973:
7970:
7962:
7961:
7957:
7952:
7948:
7943:
7939:
7934:
7931:
7925:
7924:
7920:
7915:
7911:
7897:
7892:
7888:
7883:
7880:
7874:
7873:
7869:
7864:
7860:
7855:
7852:
7849:
7843:
7842:
7834:
7829:
7817:
7812:
7809:
7806:
7798:
7797:
7785:
7780:
7776:
7771:
7768:
7762:
7761:
7757:
7752:
7749:
7746:
7740:
7739:
7727:
7722:
7719:
7716:
7692:
7691:
7652:
7616:
7601:
7600:
7568:
7544:
7535:
7526:
7519:
7510:
7484:
7483:
7469:
7436:
7417:
7416:
7380:
7344:
7325:
7324:
7284:
7260:
7251:
7242:
7233:
7224:
7209:
7208:
7190:
7157:
7138:
7137:
7104:
7060:
7046:
7027:
7026:
6986:
6962:
6953:
6944:
6937:
6928:
6896:
6870:
6869:
6851:
6811:
6797:
6782:
6781:
6763:
6730:
6716:
6695:
6692:
6657:
6651:
6599:
6593:
6564:
6558:
6525:
6501:
6495:
6435:
6411:
6405:
6380:
6352:
6305:
6281:
6256:
6235:Katsumi Nomizu
6228:indecomposable
6213:
6189:
6161:
6148:
6147:
6134:
6129:
6124:
6119:
6114:
6088:
6085:
6064:torsion tensor
5990:
5987:
5966:
5961:
5957:
5952:
5948:
5944:
5940:
5912:
5909:
5906:
5905:
5893:
5868:
5853:
5850:
5847:
5835:
5832:
5829:
5825:
5822:
5811:
5797:
5793:
5782:
5778:
5777:
5764:
5760:
5755:
5741:
5738:
5735:
5723:
5720:
5717:
5713:
5710:
5707:
5704:
5693:
5679:
5675:
5664:
5660:
5659:
5646:
5642:
5637:
5627:isomorphic to
5614:
5610:
5605:
5593:
5590:
5587:
5575:
5572:
5569:
5565:
5562:
5558:
5555:
5552:
5549:
5545:
5542:
5531:
5517:
5513:
5502:
5498:
5497:
5484:
5480:
5476:
5472:
5468:
5464:
5460:
5451:isomorphic to
5438:
5434:
5430:
5426:
5422:
5418:
5414:
5403:
5400:
5397:
5385:
5382:
5379:
5375:
5372:
5368:
5363:
5359:
5348:
5334:
5330:
5319:
5315:
5314:
5301:
5297:
5293:
5289:
5285:
5281:
5277:
5264:
5261:
5258:
5246:
5242:
5239:
5236:
5232:
5229:
5225:
5221:
5218:
5215:
5211:
5208:
5205:
5202:
5191:
5177:
5173:
5162:
5158:
5157:
5144:
5140:
5136:
5132:
5128:
5124:
5120:
5111:isomorphic to
5098:
5094:
5090:
5086:
5082:
5078:
5074:
5063:
5060:
5057:
5045:
5042:
5039:
5035:
5032:
5028:
5023:
5019:
5008:
4994:
4990:
4979:
4975:
4974:
4962:
4958:
4954:
4931:
4927:
4923:
4919:
4915:
4911:
4907:
4894:
4891:
4888:
4876:
4873:
4870:
4866:
4863:
4859:
4856:
4853:
4850:
4846:
4843:
4832:
4818:
4814:
4803:
4799:
4798:
4796:
4793:
4790:
4778:
4775:
4772:
4769:
4765:
4761:
4758:
4755:
4751:
4748:
4737:
4723:
4719:
4708:
4704:
4703:
4690:
4685:
4682:
4658:
4654:
4650:
4646:
4642:
4638:
4634:
4623:
4620:
4617:
4603:
4599:
4588:
4574:
4570:
4559:
4555:
4554:
4541:
4537:
4533:
4529:
4525:
4521:
4517:
4503:
4500:
4497:
4485:
4482:
4479:
4475:
4472:
4468:
4465:
4462:
4459:
4455:
4452:
4441:
4427:
4423:
4412:
4408:
4407:
4394:
4390:
4386:
4382:
4378:
4374:
4370:
4348:
4344:
4340:
4336:
4332:
4328:
4324:
4313:
4310:
4307:
4295:
4292:
4289:
4285:
4282:
4278:
4275:
4272:
4269:
4265:
4262:
4251:
4237:
4233:
4222:
4218:
4217:
4215:
4212:
4209:
4197:
4194:
4191:
4188:
4184:
4180:
4177:
4174:
4170:
4167:
4156:
4142:
4138:
4127:
4123:
4122:
4109:
4106:
4103:
4098:
4078:
4067:
4064:
4061:
4058:
4055:
4052:
4042:
4031:
4028:
4025:
4015:
4003:
4000:
3997:
3993:
3990:
3986:
3983:
3980:
3977:
3973:
3970:
3959:
3947:
3944:
3941:
3938:
3935:
3931:
3928:
3917:
3913:
3912:
3898:
3893:
3880:
3869:
3859:
3848:
3845:
3842:
3839:
3836:
3833:
3823:
3811:
3808:
3805:
3801:
3790:
3778:
3775:
3772:
3768:
3765:
3754:
3750:
3749:
3736:
3733:
3728:
3715:
3704:
3701:
3697:
3693:
3690:
3680:
3669:
3666:
3663:
3660:
3657:
3654:
3644:
3632:
3629:
3626:
3622:
3611:
3599:
3596:
3593:
3590:
3586:
3583:
3572:
3568:
3567:
3554:
3551:
3548:
3543:
3523:
3512:
3509:
3506:
3503:
3500:
3497:
3487:
3476:
3473:
3463:
3451:
3448:
3445:
3441:
3438:
3434:
3431:
3428:
3425:
3421:
3418:
3407:
3395:
3392:
3389:
3386:
3383:
3379:
3376:
3365:
3361:
3360:
3347:
3344:
3341:
3336:
3316:
3305:
3302:
3299:
3296:
3293:
3290:
3280:
3269:
3266:
3263:
3253:
3241:
3238:
3235:
3232:
3228:
3224:
3221:
3218:
3215:
3211:
3207:
3203:
3192:
3180:
3177:
3174:
3171:
3168:
3164:
3161:
3150:
3146:
3145:
3131:
3128:
3123:
3110:
3099:
3096:
3093:
3083:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3032:
3020:
3017:
3014:
3010:
3007:
2996:
2984:
2981:
2978:
2975:
2971:
2968:
2957:
2953:
2952:
2938:
2933:
2920:
2909:
2906:
2903:
2893:
2882:
2878:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2837:
2825:
2822:
2819:
2815:
2812:
2801:
2789:
2786:
2783:
2779:
2776:
2765:
2761:
2760:
2757:
2754:
2751:
2746:
2741:
2715:
2712:
2691:that contains
2673:
2666:
2659:
2652:
2645:
2561:
2560:
2550:
2536:
2525:Euclidean type
2505:
2502:
2439:Main article:
2436:
2433:
2391:
2379:
2373:
2356:
2347:
2329:
2328:
2317:
2314:
2310:
2307:
2301:
2298:
2294:
2290:
2287:
2284:
2281:
2278:
2274:
2271:
2266:
2263:
2260:
2257:
2254:
2249:
2245:
2183:, we obtain a
2143:To summarize,
2129:
2123:
2119:
2113:
2109:
2105:
2083:
2079:
2055:
2054:
2041:
2037:
2033:
2030:
2027:
2022:
2018:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1968:
1956:
1939:
1916:isotropy group
1867:
1864:
1849:
1825:
1797:
1770:
1759:brackets into
1746:
1718:
1694:
1670:
1646:
1622:
1598:
1585:
1584:
1573:
1568:
1563:
1560:
1555:
1550:
1545:
1540:
1536:
1531:
1526:
1523:
1518:
1513:
1508:
1503:
1499:
1494:
1489:
1486:
1481:
1476:
1471:
1466:
1452:
1451:
1438:
1433:
1428:
1423:
1418:
1404:decomposition
1387:
1359:
1327:
1291:
1259:
1258:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1203:
1199:
1177:
1122:
1119:
1099:
1096:
1093:
1090:
1087:
1084:
1064:Every compact
1038:
1035:
988:if and only if
975:
972:
958:is said to be
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
854:
851:
848:
845:
842:
839:
795:
792:
730:
725:
721:
623:
622:
620:
619:
612:
605:
597:
594:
593:
592:
591:
586:
578:
577:
573:
572:
567:
565:Harish-Chandra
562:
557:
552:
547:
542:
540:Henri Poincaré
537:
531:
530:
527:
526:
523:
522:
518:
517:
512:
507:
502:
496:
495:
490:Lie groups in
489:
488:
485:
484:
480:
479:
474:
469:
464:
459:
454:
448:
447:
442:
441:
438:
437:
433:
432:
427:
422:
420:
419:
414:
408:
406:
404:
403:
398:
392:
390:
385:
379:
378:
373:
372:
369:
368:
364:
363:
358:
353:
348:
346:
345:
340:
334:
332:
327:
322:
316:
315:
310:
309:
306:
305:
301:
300:
295:
290:
288:Diffeomorphism
285:
280:
275:
270:
264:
263:
258:
257:
254:
253:
248:
247:
246:
245:
241:
237:
233:
229:
225:
221:
217:
213:
209:
202:
201:
197:
196:
195:
194:
188:
184:
178:
174:
168:
164:
158:
151:
150:
144:
139:
138:
135:
134:
130:
129:
119:
109:
99:
89:
79:
72:Special linear
69:
62:General linear
58:
57:
52:
51:
48:
47:
39:
38:
17:
9:
6:
4:
3:
2:
11070:
11059:
11056:
11054:
11051:
11049:
11046:
11044:
11041:
11040:
11038:
11028:
11022:
11018:
11013:
11009:
11004:
11000:
10996:
10992:
10991:Selberg, Atle
10988:
10985:
10981:
10977:
10973:
10969:
10965:
10961:
10957:
10953:
10949:
10944:
10940:
10935:
10931:
10929:0-471-15732-5
10925:
10921:
10920:
10914:
10910:
10906:
10903:(1): 79–106.
10902:
10898:
10891:
10886:
10883:
10881:0-12-338301-3
10877:
10872:
10871:
10864:
10860:
10858:0-12-338460-5
10854:
10850:
10845:
10842:
10836:
10832:
10827:
10823:
10818:
10814:
10810:
10805:
10801:
10796:
10792:
10788:
10784:
10780:
10777:
10775:0-8218-0288-7
10771:
10767:
10763:
10762:Borel, Armand
10759:
10755:
10753:0-387-15279-2
10749:
10744:
10743:
10737:
10733:
10729:
10724:
10721:(2): 85–177,
10720:
10716:
10712:
10708:
10705:
10699:
10695:
10690:
10687:
10683:
10679:
10675:
10671:
10666:
10665:
10659:
10653:
10649:
10639:
10636:
10634:
10631:
10629:
10626:
10624:
10621:
10620:
10614:
10612:
10608:
10604:
10598:
10588:
10586:
10582:
10578:
10574:
10570:
10566:
10562:
10558:
10554:
10550:
10546:
10541:
10539:
10535:
10529:
10519:
10515:
10509:
10505:
10501:
10497:
10492:
10490:
10484:
10474:
10472:
10468:
10462:
10447:
10444:
10442:
10438:
10434:
10411:
10397:
10394:
10383:
10379:
10375:
10370:
10367:
10364:
10357:
10343:
10340:
10329:
10325:
10321:
10316:
10310:
10307:
10304:
10294:
10293:
10292:
10254:
10251:
10248:
10225:
10222:
10214:
10202:
10197:
10178:
10177:
10176:
10157:
10145:
10142:
10139:
10134:
10122:
10108:
10107:
10106:
10060:
10056:
10052:
10047:
10043:
10034:
10030:
10026:
10018:
10014:
10010:
10000:
9996:
9989:
9981:
9977:
9973:
9968:
9964:
9957:
9954:
9946:
9942:
9938:
9933:
9929:
9920:
9916:
9912:
9904:
9900:
9896:
9886:
9882:
9871:
9870:
9869:
9850:
9846:
9840:
9836:
9832:
9822:
9818:
9810:
9809:
9808:
9768:
9764:
9760:
9757:
9754:
9749:
9745:
9736:
9717:
9714:
9711:
9705:
9658:
9655:
9652:
9623:
9620:
9617:
9611:
9608:
9596:
9592:
9589:
9579:
9578:
9577:
9557:
9550:
9481:Factorization
9478:
9452:
9449:
9446:
9423:
9405:
9386:
9383:
9380:
9377:
9374:
9371:
9368:
9362:
9359:
9356:
9350:
9347:
9344:
9338:
9318:
9313:
9309:
9286:
9278:
9275:
9272:
9230:
9216:
9213:
9210:
9207:
9198:
9195:
9192:
9186:
9183:
9169:
9166:
9161:
9157:
9153:
9150:
9147:
9144:
9136:
9128:
9125:
9122:
9113:
9108:
9093:
9090:
9087:
9077:
9076:
9075:
9073:
9069:
9065:
9061:
9060:metric tensor
9051:
9043:
9041:
9037:
9032:
9030:
9026:
9022:
9014:
9010:
9006:
9002:
8998:
8995:
8991:
8988:
8987:
8986:
8984:
8980:
8976:
8972:
8968:
8964:
8960:
8956:
8951:
8949:
8945:
8941:
8937:
8933:
8932:Gelfand pairs
8929:
8925:
8921:
8917:
8913:
8909:
8905:
8901:
8897:
8893:
8889:
8885:
8881:
8877:
8873:
8869:
8865:
8861:
8857:
8854:In the 1950s
8851:
8836:
8816:
8813:
8801:
8790:
8784:
8783:
8771:
8759:
8756:
8744:
8737:
8731:
8730:
8718:
8711:
8708:
8702:
8701:
8697:
8685:
8682:
8674:
8671:
8665:
8664:
8638:
8636:)×Sp(1)
8635:
8625:
8617:
8608:
8600:
8598:× SO(2)
8589:
8583:
8582:
8560:
8557:
8553:
8539:
8526:
8519:
8513:
8512:
8490:
8488:)×Sp(1)
8487:
8477:
8469:
8466:
8456:
8442:
8435:
8429:
8428:
8425:× SO(2)
8416:
8409:
8402:
8399:
8393:
8392:
8388:
8376:
8373:
8369:
8361:
8358:
8350:
8347:
8341:
8340:
8325:
8318:
8316:)×Sp(1)
8315:
8307:
8300:
8290:
8284:
8283:
8272:
8270:)×SO(2)
8269:
8259:
8256:
8242:
8235:
8228:
8222:
8221:
8210:
8208:)×SO(2)
8207:
8193:
8190:
8176:
8173:
8159:
8152:
8146:
8145:
8134:
8127:
8125:)×SU(2)
8124:
8114:
8110:
8102:
8099:
8085:
8078:
8072:
8071:
8060:
8053:
8046:
8039:
8036:
8030:
8029:
8018:
8015:
8011:
8003:
8000:
7996:
7988:
7985:
7977:
7974:
7968:
7967:
7956:
7953:
7947:
7944:
7938:
7935:
7930:
7927:
7926:
7919:
7916:
7907:
7903:
7896:
7893:
7887:
7884:
7879:
7876:
7875:
7868:
7865:
7859:
7856:
7853:
7848:
7845:
7844:
7840:
7833:
7830:
7827:
7823:
7816:
7813:
7810:
7804:
7803:
7795:
7792:)× SL(2,
7791:
7784:
7781:
7775:
7772:
7767:
7764:
7763:
7756:
7753:
7750:
7745:
7742:
7741:
7737:
7734:)× SL(2,
7733:
7726:
7723:
7720:
7715:
7712:
7711:
7708:
7704:
7699:
7689:
7685:
7681:
7676:
7673: =
7672:
7669: +
7668:
7664:
7660:
7656:
7653:
7650:
7646:
7642:
7636:
7632:
7628:
7624:
7620:
7617:
7614:
7610:
7606:
7603:
7602:
7598:
7594:
7590:
7586:
7580:
7576:
7572:
7569:
7566:
7562:
7558:
7554:
7547:
7543:
7538:
7534:
7529:
7525:
7518:
7514:
7511:
7509:
7506: =
7505:
7502: +
7501:
7497:
7493:
7489:
7486:
7485:
7481:
7477:
7473:
7470:
7468:
7465: =
7464:
7461: +
7460:
7456:
7452:
7448:
7444:
7440:
7437:
7434:
7430:
7426:
7423:
7422:
7414:
7410:
7406:
7400:
7396:
7392:
7388:
7384:
7381:
7378:
7374:
7370:
7364:
7360:
7356:
7352:
7348:
7345:
7342:
7338:
7334:
7330:
7327:
7326:
7322:
7318:
7314:
7310:
7304:
7300:
7296:
7292:
7288:
7285:
7282:
7278:
7274:
7270:
7263:
7259:
7254:
7250:
7245:
7241:
7236:
7232:
7228:
7225:
7222:
7218:
7214:
7211:
7210:
7206:
7202:
7198:
7194:
7191:
7189:
7185:
7181:
7177:
7173:
7169:
7165:
7161:
7158:
7155:
7151:
7147:
7144:
7143:
7136:
7132:
7128:
7124:
7120:
7116:
7112:
7108:
7105:
7102:
7098:
7094:
7088:
7084:
7080:
7076:
7072:
7068:
7064:
7061:
7058:
7054:
7050:
7047:
7044:
7040:
7036:
7032:
7029:
7028:
7024:
7020:
7016:
7012:
7006:
7002:
6998:
6994:
6990:
6987:
6984:
6980:
6976:
6972:
6965:
6961:
6956:
6952:
6947:
6943:
6936:
6932:
6929:
6926:
6922:
6918:
6914:
6908:
6904:
6900:
6897:
6895:
6891:
6887:
6883:
6879:
6875:
6872:
6871:
6867:
6863:
6859:
6855:
6852:
6849:
6845:
6841:
6837:
6831:
6827:
6823:
6819:
6815:
6812:
6809:
6805:
6801:
6798:
6795:
6791:
6787:
6784:
6783:
6779:
6775:
6771:
6767:
6764:
6762:
6758:
6754:
6750:
6746:
6742:
6738:
6734:
6731:
6728:
6724:
6720:
6717:
6714:
6710:
6706:
6703:
6702:
6699:
6691:
6689:
6685:
6681:
6677:
6673:
6655:
6638:
6634:
6630:
6625:
6623:
6619:
6615:
6597:
6580:
6562:
6545:
6541:
6499:
6481:
6479:
6475:
6471:
6467:
6463:
6459:
6455:
6451:
6409:
6368:
6340:
6336:
6332:
6327:
6325:
6321:
6244:
6240:
6236:
6231:
6229:
6177:
6127:
6117:
6103:
6102:
6101:
6099:
6095:
6084:
6081:
6077:
6073:
6069:
6066:vanishes) on
6065:
6061:
6057:
6052:
6048:
6044:
6038:
6036:
6032:
6028:
6024:
6020:
6016:
6012:
6008:
6004:
6000:
5996:
5986:
5984:
5980:
5964:
5959:
5946:
5930:
5926:
5922:
5918:
5883:
5858:
5854:
5851:
5848:
5830:
5812:
5795:
5791:
5783:
5780:
5779:
5762:
5758:
5745:
5742:
5739:
5736:
5718:
5694:
5677:
5673:
5665:
5662:
5661:
5644:
5640:
5612:
5608:
5594:
5591:
5588:
5570:
5556:
5550:
5532:
5515:
5511:
5503:
5500:
5499:
5482:
5478:
5466:
5436:
5432:
5420:
5404:
5401:
5398:
5380:
5366:
5361:
5357:
5349:
5332:
5328:
5320:
5317:
5316:
5299:
5295:
5283:
5268:
5265:
5262:
5259:
5230:
5223:
5216:
5192:
5175:
5171:
5163:
5160:
5159:
5142:
5138:
5126:
5096:
5092:
5080:
5064:
5061:
5058:
5040:
5026:
5021:
5017:
5009:
4992:
4988:
4980:
4977:
4976:
4956:
4929:
4925:
4913:
4898:
4895:
4892:
4889:
4871:
4857:
4851:
4833:
4816:
4812:
4804:
4801:
4800:
4797:
4794:
4791:
4773:
4770:
4763:
4756:
4738:
4721:
4717:
4709:
4706:
4705:
4688:
4671:isometric to
4656:
4652:
4640:
4624:
4621:
4618:
4601:
4597:
4589:
4572:
4568:
4560:
4557:
4556:
4539:
4535:
4523:
4508:
4505:Complexified
4504:
4501:
4498:
4480:
4466:
4460:
4442:
4425:
4421:
4413:
4410:
4409:
4392:
4388:
4376:
4361:isometric to
4346:
4342:
4330:
4314:
4311:
4308:
4290:
4276:
4270:
4252:
4235:
4231:
4223:
4220:
4219:
4216:
4213:
4210:
4192:
4189:
4182:
4175:
4157:
4140:
4136:
4128:
4125:
4124:
4107:
4104:
4101:
4086:
4082:
4079:
4062:
4059:
4056:
4043:
4029:
4026:
4023:
4016:
3998:
3984:
3978:
3960:
3942:
3939:
3936:
3918:
3915:
3914:
3896:
3881:
3867:
3860:
3843:
3840:
3837:
3831:
3824:
3806:
3791:
3773:
3755:
3752:
3751:
3734:
3731:
3716:
3699:
3695:
3691:
3681:
3664:
3661:
3658:
3652:
3645:
3627:
3612:
3594:
3591:
3573:
3570:
3569:
3552:
3549:
3546:
3531:
3527:
3524:
3507:
3504:
3501:
3488:
3474:
3471:
3464:
3446:
3432:
3426:
3408:
3390:
3387:
3384:
3366:
3363:
3362:
3345:
3342:
3339:
3324:
3320:
3317:
3300:
3297:
3294:
3281:
3267:
3264:
3261:
3254:
3233:
3222:
3216:
3193:
3175:
3172:
3169:
3151:
3148:
3147:
3129:
3126:
3111:
3097:
3094:
3091:
3084:
3067:
3064:
3061:
3058:
3049:
3046:
3043:
3033:
3015:
2997:
2979:
2976:
2958:
2955:
2954:
2936:
2921:
2907:
2904:
2901:
2894:
2880:
2876:
2869:
2866:
2863:
2854:
2851:
2848:
2820:
2802:
2784:
2766:
2763:
2762:
2758:
2755:
2752:
2750:
2747:
2745:
2742:
2739:
2738:
2735:
2733:
2729:
2725:
2721:
2711:
2708:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2677:
2672:
2665:
2658:
2651:
2644:
2641:
2637:
2633:
2629:
2624:
2622:
2621:
2615:
2611:
2607:
2603:
2599:
2595:
2591:
2587:
2582:
2580:
2575:
2573:
2568:
2566:
2558:
2554:
2551:
2548:
2544:
2540:
2537:
2534:
2530:
2526:
2523:
2522:
2521:
2519:
2514:
2511:
2501:
2499:
2495:
2491:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2459:
2455:
2450:
2448:
2442:
2432:
2430:
2426:
2422:
2418:
2414:
2410:
2405:
2403:
2399:
2394:
2390:
2386:
2382:
2376:
2372:
2368:
2364:
2359:
2355:
2350:
2346:
2342:
2338:
2334:
2315:
2308:
2305:
2299:
2296:
2292:
2285:
2282:
2276:
2272:
2269:
2264:
2261:
2255:
2252:
2247:
2243:
2235:
2234:
2233:
2232:) and define
2231:
2227:
2223:
2218:
2214:
2209:
2205:
2202:To show that
2200:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2141:
2127:
2121:
2111:
2107:
2081:
2077:
2068:
2064:
2060:
2039:
2035:
2031:
2028:
2025:
2020:
2016:
2009:
2006:
2003:
1997:
1994:
1991:
1984:
1983:
1982:
1980:
1976:
1971:
1967:
1963:
1959:
1954:
1950:
1946:
1942:
1937:
1933:
1929:
1925:
1921:
1917:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1863:
1813:
1784:
1734:
1571:
1561:
1548:
1534:
1524:
1511:
1497:
1487:
1474:
1457:
1456:
1455:
1431:
1421:
1407:
1406:
1405:
1403:
1375:
1347:
1343:
1315:
1311:
1307:
1279:
1275:
1270:
1268:
1264:
1245:
1239:
1236:
1230:
1224:
1221:
1218:
1215:
1212:
1206:
1201:
1197:
1189:
1188:
1187:
1185:
1180:
1175:
1171:
1167:
1163:
1159:
1156:
1152:
1148:
1144:
1140:
1136:
1132:
1128:
1118:
1116:
1111:
1094:
1091:
1088:
1082:
1074:
1069:
1067:
1062:
1060:
1056:
1052:
1048:
1044:
1034:
1030:
1028:
1024:
1020:
1016:
1012:
1008:
1003:
1001:
997:
993:
989:
985:
982:implies that
981:
971:
969:
965:
961:
957:
953:
951:
947:
943:
939:
935:
934:tangent space
931:
927:
911:
905:
902:
896:
893:
884:
878:
872:
852:
849:
843:
837:
829:
825:
821:
817:
813:
809:
805:
801:
791:
789:
788:Marcel Berger
784:
782:
778:
774:
770:
766:
763:
759:
755:
750:
748:
744:
728:
723:
719:
710:
706:
702:
698:
694:
690:
684:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
642:
638:
634:
630:
618:
613:
611:
606:
604:
599:
598:
596:
595:
590:
587:
585:
582:
581:
580:
579:
571:
568:
566:
563:
561:
558:
556:
553:
551:
548:
546:
543:
541:
538:
536:
533:
532:
525:
524:
516:
513:
511:
508:
506:
503:
501:
498:
497:
493:
487:
486:
478:
475:
473:
470:
468:
465:
463:
460:
458:
455:
453:
450:
449:
445:
440:
439:
431:
428:
426:
423:
418:
415:
413:
410:
409:
407:
402:
399:
397:
394:
393:
391:
389:
386:
384:
381:
380:
376:
371:
370:
362:
359:
357:
354:
352:
349:
344:
341:
339:
336:
335:
333:
331:
328:
326:
323:
321:
318:
317:
313:
308:
307:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
269:
266:
265:
261:
256:
255:
244:
238:
236:
230:
228:
222:
220:
214:
212:
206:
205:
204:
203:
199:
198:
193:
191:
185:
183:
181:
175:
173:
171:
165:
163:
161:
155:
154:
153:
152:
148:
147:
142:
137:
136:
127:
123:
120:
117:
113:
110:
107:
103:
100:
97:
93:
90:
87:
83:
80:
77:
73:
70:
67:
63:
60:
59:
55:
50:
49:
45:
41:
40:
37:
33:
30:
29:
24:
16:
11016:
11007:
10998:
10994:
10970:(1): 33–65,
10967:
10963:
10947:
10938:
10918:
10900:
10896:
10869:
10848:
10830:
10812:
10808:
10790:
10786:
10783:Cartan, Élie
10765:
10741:
10718:
10714:
10693:
10677:
10673:
10657:
10652:
10600:
10584:
10580:
10576:
10572:
10568:
10564:
10552:
10548:
10544:
10542:
10533:
10531:
10513:
10507:
10503:
10499:
10495:
10493:
10486:
10464:
10445:
10436:
10432:
10430:
10240:
10174:
10080:
9867:
9734:
9641:
9484:
9404:Killing form
9261:
9072:Killing form
9067:
9063:
9057:
9049:
9033:
9028:
9024:
9020:
9018:
9012:
9008:
9004:
9000:
8993:
8989:
8985:, such that
8982:
8978:
8974:
8970:
8966:
8962:
8958:
8954:
8952:
8947:
8943:
8939:
8923:
8919:
8915:
8911:
8907:
8903:
8899:
8895:
8891:
8887:
8883:
8879:
8878:normalising
8875:
8871:
8867:
8863:
8859:
8856:Atle Selberg
8853:
8834:
8811:
8799:×Sp(1)
8780:×SU(2)
8769:
8754:
8727:×Sp(1)
8695:
8680:
8647:×SO(2)
8633:
8623:
8606:
8579:×SO(2)
8569:×SO(2)
8555:
8554:)×SL(2,
8551:
8509:×SO(2)
8485:
8475:
8464:
8454:
8386:
8371:
8370:)×Sp(2,
8367:
8356:
8332:4(−20)
8313:
8267:
8254:
8205:
8188:
8171:
8122:
8112:
8111:)×SL(2,
8108:
8097:
8013:
8012:)×SO(2,
8009:
7998:
7997:)×SL(2,
7994:
7983:
7954:
7945:
7936:
7928:
7917:
7905:
7904:)×Sp(2,
7901:
7894:
7885:
7877:
7866:
7857:
7846:
7838:
7831:
7825:
7824:)×Sp(2,
7821:
7814:
7793:
7789:
7782:
7773:
7765:
7754:
7743:
7735:
7731:
7724:
7713:
7702:
7697:
7695:
7687:
7683:
7679:
7674:
7670:
7666:
7662:
7658:
7654:
7648:
7644:
7640:
7634:
7630:
7626:
7622:
7618:
7612:
7608:
7604:
7596:
7592:
7588:
7584:
7578:
7574:
7570:
7564:
7560:
7556:
7552:
7545:
7541:
7536:
7532:
7527:
7523:
7516:
7512:
7507:
7503:
7499:
7495:
7491:
7487:
7479:
7475:
7471:
7466:
7462:
7458:
7454:
7450:
7446:
7442:
7438:
7432:
7428:
7424:
7412:
7408:
7404:
7398:
7394:
7390:
7386:
7382:
7376:
7372:
7368:
7362:
7358:
7354:
7350:
7346:
7340:
7336:
7332:
7328:
7320:
7316:
7312:
7308:
7302:
7298:
7294:
7290:
7286:
7280:
7276:
7272:
7268:
7261:
7257:
7252:
7248:
7243:
7239:
7234:
7230:
7226:
7220:
7216:
7212:
7204:
7200:
7196:
7192:
7187:
7183:
7179:
7175:
7171:
7167:
7163:
7159:
7153:
7149:
7145:
7134:
7130:
7126:
7122:
7118:
7114:
7110:
7106:
7100:
7096:
7092:
7086:
7082:
7078:
7074:
7070:
7066:
7062:
7056:
7052:
7048:
7042:
7038:
7034:
7030:
7022:
7018:
7014:
7010:
7004:
7000:
6996:
6992:
6988:
6982:
6978:
6974:
6970:
6963:
6959:
6954:
6950:
6945:
6941:
6934:
6930:
6924:
6920:
6916:
6912:
6906:
6902:
6898:
6893:
6889:
6885:
6881:
6877:
6873:
6865:
6861:
6857:
6853:
6847:
6843:
6839:
6835:
6829:
6825:
6821:
6817:
6813:
6807:
6803:
6799:
6793:
6789:
6785:
6777:
6773:
6769:
6765:
6760:
6756:
6752:
6748:
6744:
6740:
6736:
6732:
6726:
6722:
6718:
6712:
6708:
6704:
6697:
6683:
6679:
6675:
6671:
6636:
6632:
6628:
6626:
6621:
6617:
6613:
6578:
6543:
6482:
6477:
6473:
6469:
6465:
6461:
6457:
6453:
6449:
6366:
6338:
6334:
6330:
6328:
6323:
6319:
6242:
6238:
6232:
6149:
6097:
6093:
6090:
6067:
6055:
6050:
6046:
6042:
6039:
6034:
6030:
6010:
6006:
6002:
5994:
5992:
5982:
5978:
5914:
4084:
4081:Grassmannian
3529:
3526:Grassmannian
3322:
3319:Grassmannian
2748:
2743:
2731:
2727:
2723:
2719:
2717:
2709:
2704:
2700:
2696:
2692:
2688:
2684:
2680:
2678:
2670:
2663:
2656:
2649:
2642:
2635:
2631:
2627:
2625:
2619:
2613:
2609:
2605:
2601:
2597:
2593:
2589:
2583:
2578:
2576:
2571:
2569:
2564:
2562:
2556:
2552:
2542:
2539:Compact type
2538:
2528:
2524:
2517:
2515:
2509:
2507:
2497:
2489:
2485:
2481:
2473:
2469:
2465:
2461:
2457:
2453:
2451:
2444:
2428:
2424:
2420:
2416:
2412:
2408:
2406:
2401:
2397:
2392:
2388:
2384:
2380:
2374:
2370:
2366:
2362:
2357:
2353:
2348:
2344:
2340:
2336:
2332:
2330:
2229:
2225:
2221:
2220:(a coset of
2216:
2212:
2207:
2203:
2201:
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2164:
2160:
2156:
2152:
2148:
2144:
2142:
2066:
2063:automorphism
2056:
1978:
1974:
1969:
1965:
1961:
1957:
1952:
1944:
1940:
1935:
1931:
1927:
1923:
1919:
1914:denotes the
1911:
1907:
1903:
1899:
1895:
1891:
1887:
1879:
1875:
1871:
1869:
1811:
1785:
1586:
1453:
1373:
1345:
1341:
1313:
1309:
1305:
1277:
1273:
1271:
1266:
1262:
1260:
1183:
1178:
1173:
1169:
1165:
1161:
1157:
1150:
1146:
1142:
1138:
1134:
1126:
1124:
1112:
1070:
1063:
1040:
1031:
1026:
1022:
1018:
1014:
1006:
1004:
983:
977:
967:
963:
959:
955:
954:
949:
945:
941:
937:
929:
925:
827:
823:
819:
815:
811:
807:
803:
799:
797:
785:
783:is compact.
780:
776:
768:
764:
757:
753:
751:
708:
704:
700:
696:
692:
688:
685:
651:contains an
632:
626:
570:Armand Borel
555:Hermann Weyl
356:Loop algebra
338:Killing form
312:Lie algebras
189:
179:
169:
159:
125:
115:
105:
95:
85:
75:
65:
36:Lie algebras
15:
10815:: 114–134,
10793:: 214–216,
10607:loop spaces
9436:this makes
9040:Wolf (2007)
8833:×SL(2,
8768:×SL(2,
8694:×Sp(2,
8385:×SO(2,
7629:)×Sp(2
7449:)×Sp(2
6542:involution
6318:is simple,
3321:of complex
2510:irreducible
2447:Élie Cartan
1011:homogeneous
806:a point of
629:mathematics
550:Élie Cartan
396:Root system
200:Exceptional
11053:Lie groups
11037:Categories
10956:Nomizu, K.
10950:, Benjamin
10941:, Benjamin
10644:References
10471:7 families
9046:Properties
8902:such that
7531:)×Sp(
7247:)×SO(
7170:)×SO(
7073:)×GL(
6824:)×GL(
6743:)×GL(
6540:antilinear
2753:Dimension
2496:, because
2061:Lie group
2059:involutive
1981:, the map
1890:(that is,
1402:direct sum
1155:involution
1073:lens space
1059:Lie groups
948:to all of
773:involution
665:Lie theory
649:isometries
535:Sophus Lie
528:Scientists
401:Weyl group
122:Symplectic
82:Orthogonal
32:Lie groups
10380:λ
10368:⋯
10326:λ
10314:⟩
10311:⋅
10305:⋅
10302:⟨
10252:≠
10146:⊕
10143:⋯
10140:⊕
10066:⟩
10040:⟨
10031:λ
10024:⟩
10006:#
9993:⟨
9952:⟩
9926:⟨
9917:λ
9910:⟩
9892:#
9879:⟨
9837:λ
9828:#
9758:…
9718:⋅
9712:⋅
9662:⟩
9659:⋅
9653:⋅
9650:⟨
9606:⟩
9601:#
9587:⟨
9562:#
9554:↦
9524:→
9457:⟩
9453:⋅
9447:⋅
9444:⟨
9384:
9378:∘
9372:
9363:
9283:⟩
9279:⋅
9273:⋅
9270:⟨
9238:otherwise
9217:∈
9184:−
9170:≅
9154:∈
9133:⟩
9120:⟨
9098:⟩
9085:⟨
6949:)×U(
6202:. Since
6128:⊕
6072:curvature
5947:⊗
5557:⋅
5467:⊗
5421:⊗
5367:⋅
5284:⊗
5231:±
5127:⊗
5081:⊗
5027:⋅
4957:⊗
4914:⊗
4858:⋅
4771:±
4641:⊗
4524:⊗
4467:⋅
4377:⊗
4331:⊗
4277:⋅
4190:±
3985:×
3662:−
3433:×
3223:×
3095:−
3047:−
2905:−
2852:−
2297:−
2286:σ
2280:↦
2259:→
2112:σ
2082:σ
2032:∘
2026:∘
2013:↦
2001:→
1992:σ
1884:Lie group
1562:⊂
1525:⊂
1488:⊂
1432:⊕
1225:σ
1216:∈
1202:σ
1133:. Then a
1131:Lie group
1025:(because
903:−
897:γ
879:γ
838:γ
762:Lie group
412:Real form
298:Euclidean
149:Classical
10958:(1954),
10764:(2001),
10738:(1987),
10680:: 3–24,
10617:See also
8709:–
8672:–
8400:–
8348:–
8266:or Sk(5,
8037:–
7975:–
7854:–
7811:–
7751:–
7721:–
6577:, while
6456:, where
6076:parallel
6005:, i.e.,
2618:rank of
2309:′
2273:′
2224:, where
1910:, where
1372:. Since
1261:Because
1164:). Thus
1037:Examples
1000:complete
743:complete
661:holonomy
643:) whose
584:Glossary
278:Poincaré
11001:: 47–87
10984:2372398
10601:In the
9543:taking
9402:is the
6337:×
5927:, or a
2387:. Thus
2339:fixing
1160:in Aut(
1047:spheres
707:fixing
492:physics
273:Lorentz
102:Unitary
11023:
10982:
10926:
10878:
10855:
10837:
10772:
10750:
10700:
10605:, the
9642:where
9421:
9331:, and
9262:Here,
9007:sends
8992:fixes
8831:7(−25)
8827:8(−24)
8819:8(−24)
8808:8(−24)
8804:8(−24)
8793:8(−24)
8787:8(−24)
8772:) or E
8655:6(−26)
8651:7(−25)
8645:6(−14)
8641:7(−25)
8630:7(−25)
8620:7(−25)
8613:7(−25)
8603:7(−25)
8592:7(−25)
8586:7(−25)
8577:6(−14)
8328:6(−26)
8321:6(−26)
8310:6(−26)
8303:6(−26)
8293:6(−26)
8287:6(−26)
8279:4(−20)
8275:6(−14)
8262:6(−14)
8251:6(−14)
8245:6(−14)
8238:6(−14)
8231:6(−14)
8225:6(−14)
7958:4(−20)
7949:4(−20)
7940:4(−20)
7932:4(−20)
7583:or Sp(
7551:or Sp(
7307:or SO(
7267:or SO(
7125:even,
7009:or SU(
6969:or SU(
6911:or SU(
6694:Tables
6174:is an
6070:whose
5161:EVIII
2740:Label
2634:), Sp(
2630:), SU(
2600:×
2331:where
2057:is an
1659:is an
1071:Every
1053:, and
669:Cartan
268:Circle
10980:JSTOR
10893:(PDF)
10559:on a
10175:with
9807:with
9360:trace
9019:When
8823:7(−5)
8778:7(−5)
8573:7(−5)
8563:7(−5)
8548:7(−5)
8542:7(−5)
8535:7(−5)
8529:7(−5)
8522:7(−5)
8516:7(−5)
7401:even
7393:/2),
7365:even
7357:/2),
7343:even
7305:even
7297:/2),
7207:even
7117:/2),
7089:even
7045:even
7033:= SL(
7007:even
6999:/2),
6876:= SU(
6868:even
6850:even
6788:= SL(
6780:even
6707:= SL(
6686:is a
6472:with
6393:. If
4978:EVII
4944:over
4411:EIII
3571:DIII
3149:AIII
2756:Rank
2456:let (
1949:1-jet
1882:is a
1454:with
1172:with
865:then
645:group
635:is a
343:Index
11021:ISBN
10924:ISBN
10876:ISBN
10853:ISBN
10835:ISBN
10770:ISBN
10748:ISBN
10698:ISBN
10437:i.e.
10433:e.g.
10241:for
9698:and
9058:The
9011:to −
8981:and
8910:and
8774:8(8)
8766:7(7)
8762:8(8)
8751:8(8)
8747:8(8)
8740:8(8)
8734:8(8)
8649:or E
8628:or E
8611:or E
8571:or E
8567:6(2)
8546:or E
8533:or E
8507:6(2)
8503:7(7)
8501:or E
8497:6(6)
8493:7(7)
8482:7(7)
8480:or E
8472:7(7)
8461:7(7)
8459:or E
8451:7(7)
8449:or E
8445:7(7)
8438:7(7)
8432:7(7)
8249:or E
8217:4(4)
8213:6(2)
8202:6(2)
8200:or E
8196:6(2)
8185:6(2)
8183:or E
8179:6(2)
8168:6(2)
8166:or E
8162:6(2)
8155:6(2)
8149:6(2)
8141:4(4)
8137:6(6)
8130:6(6)
8119:6(6)
8117:or E
8105:6(6)
8094:6(6)
8092:or E
8088:6(6)
8081:6(6)
8075:6(6)
7921:4(4)
7912:4(4)
7910:or F
7898:4(4)
7889:4(4)
7881:4(4)
7786:2(2)
7777:2(2)
7769:2(2)
7215:=SO(
7148:=SO(
7081:)),
6751:)),
6674:and
6025:and
5981:and
5663:FII
5399:112
5318:EIX
5260:128
4802:EVI
4558:EIV
4221:EII
3916:CII
3364:BDI
2956:AII
2730:and
2604:and
2565:rank
2365:) =
1938:on T
1182:and
1176:= id
1137:for
1125:Let
978:The
798:Let
679:and
631:, a
293:Loop
34:and
10972:doi
10905:doi
10901:350
10817:doi
10795:doi
10723:doi
10682:doi
10516:= 2
9783:of
9576:as
9003:at
8973:of
8965:at
8957:in
8942:on
8938:of
8898:in
8890:in
7705:= 0
7678:or
7665:),
7639:or
7498:),
7457:),
7411:/4,
7403:or
7389:/2,
7375:/2,
7367:or
7353:/2,
7339:),
7335:/2,
7293:/2,
7203:),
7199:/2,
7178:),
7113:/2,
7099:/2,
7091:or
7077:/2,
7069:/2,
7055:/2,
7041:),
7037:/2,
6995:/2,
6967:))
6884:),
6864:),
6846:),
6842:/2,
6834:or
6832:))
6776:),
6682:or
6546:of
6178:of
6074:is
5737:16
5589:28
5501:FI
5059:54
4890:64
4792:70
4707:EV
4619:26
4499:32
4309:40
4211:42
4126:EI
4051:min
3753:CI
3496:min
3289:min
2764:AI
2596:is
2577:B.
2570:A.
2191:on
1977:at
1926:at
1922:on
1908:G/K
1898:of
1870:If
1707:in
1340:of
1304:of
936:of
928:at
775:of
699:of
647:of
627:In
124:Sp(
114:SU(
94:SO(
74:SL(
64:GL(
11039::
10999:20
10997:,
10978:,
10968:76
10966:,
10962:,
10899:.
10895:.
10813:55
10811:,
10791:54
10789:,
10719:74
10717:,
10676:,
10540:.
10473:.
9381:ad
9369:ad
9042:.
9027:,
8916:σx
8914:=
8912:sy
8908:σy
8906:=
8904:sx
8886:,
8837:)
8814:)
8757:)
8698:)
8683:)
8626:)
8609:)
8558:)
8478:)
8467:)
8389:)
8374:)
8359:)
8257:)
8191:)
8174:)
8115:)
8100:)
8016:)
8001:)
7986:)
7841:)
7828:)
7796:)
7738:)
7707:.
7703:kl
7690:)
7651:)
7637:)
7615:)
7599:)
7581:)
7567:)
7549:)
7482:)
7435:)
7415:)
7379:)
7323:)
7283:)
7265:)
7223:)
7186:=
7182:+
7156:)
7133:=
7129:+
7103:)
7059:)
7025:)
6985:)
6927:)
6909:)
6892:=
6888:+
6810:)
6796:)
6759:=
6755:+
6729:)
6715:)
6624:.
6333:=
6045:=
6021:,
5852:2
5849:8
5781:G
5740:1
5592:4
5402:4
5263:8
5217:16
5062:3
4893:4
4852:12
4795:7
4622:2
4502:2
4461:10
4312:4
4214:6
2707:.
2676:.
2669:,
2662:,
2655:,
2648:,
2623:.
2555::
2541::
2527::
2468:,
2464:,
2460:,
2431:.
2423:,
2419:,
2415:,
2228:∈
2217:hK
2215:=
2199:.
2173:eK
1902:,
1783:.
1276:,
1117:.
1049:,
1045:,
998:,
970:.
952:.
777:G.
749:.
691:,
683:.
675:,
104:U(
84:O(
10974::
10911:.
10907::
10819::
10797::
10725::
10684::
10678:4
10585:q
10581:p
10577:q
10573:p
10569:q
10565:p
10553:K
10549:K
10545:G
10534:M
10514:p
10508:i
10504:K
10500:K
10496:G
10412:d
10406:m
10398:|
10395:B
10384:d
10376:1
10371:+
10365:+
10358:1
10352:m
10344:|
10341:B
10330:1
10322:1
10317:=
10308:,
10277:g
10255:j
10249:i
10226:0
10223:=
10220:]
10215:j
10209:m
10203:,
10198:i
10192:m
10186:[
10158:d
10152:m
10135:1
10129:m
10123:=
10118:m
10091:m
10061:i
10057:Y
10053:,
10048:j
10044:Y
10035:j
10027:=
10019:i
10015:Y
10011:,
10001:j
9997:Y
9990:=
9987:)
9982:j
9978:Y
9974:,
9969:i
9965:Y
9961:(
9958:B
9955:=
9947:j
9943:Y
9939:,
9934:i
9930:Y
9921:i
9913:=
9905:j
9901:Y
9897:,
9887:i
9883:Y
9851:i
9847:Y
9841:i
9833:=
9823:i
9819:Y
9793:m
9769:n
9765:Y
9761:,
9755:,
9750:1
9746:Y
9721:)
9715:,
9709:(
9706:B
9684:m
9656:,
9627:)
9624:Y
9621:,
9618:X
9615:(
9612:B
9609:=
9597:Y
9593:,
9590:X
9558:Y
9551:Y
9529:m
9519:m
9495:m
9462:g
9450:,
9424:;
9416:h
9390:)
9387:Y
9375:X
9366:(
9357:=
9354:)
9351:Y
9348:,
9345:X
9342:(
9339:B
9319:M
9314:p
9310:T
9287:p
9276:,
9231:0
9222:h
9214:Y
9211:,
9208:X
9202:)
9199:Y
9196:,
9193:X
9190:(
9187:B
9175:m
9167:M
9162:p
9158:T
9151:Y
9148:,
9145:X
9137:p
9129:Y
9126:,
9123:X
9114:{
9109:=
9103:g
9094:Y
9091:,
9088:X
9068:G
9064:M
9029:M
9025:X
9021:s
9015:.
9013:X
9009:X
9005:x
9001:s
8996:;
8994:x
8990:s
8983:X
8979:x
8975:M
8971:s
8967:x
8963:X
8959:M
8955:x
8948:M
8946:(
8944:L
8940:G
8924:G
8920:σ
8900:G
8896:s
8892:M
8888:y
8884:x
8880:G
8876:σ
8872:G
8868:M
8835:R
8817:E
8812:H
8802:E
8797:7
8791:E
8785:E
8770:R
8760:E
8755:H
8745:E
8738:E
8732:E
8725:7
8721:8
8719:E
8714:8
8712:E
8705:8
8703:E
8696:C
8692:7
8688:8
8686:E
8681:C
8677:8
8675:E
8668:8
8666:E
8639:E
8634:H
8624:R
8618:E
8607:H
8601:E
8596:6
8590:E
8584:E
8561:E
8556:R
8552:H
8540:E
8527:E
8520:E
8514:E
8491:E
8486:H
8476:R
8470:E
8465:H
8457:)
8455:R
8443:E
8436:E
8430:E
8423:6
8419:7
8417:E
8412:7
8410:E
8405:7
8403:E
8396:7
8394:E
8387:C
8383:6
8379:7
8377:E
8372:C
8368:C
8364:7
8362:E
8357:C
8353:7
8351:E
8344:7
8342:E
8326:E
8319:E
8314:H
8308:E
8301:E
8297:4
8291:E
8285:E
8273:E
8268:H
8260:E
8255:R
8243:E
8236:E
8229:E
8223:E
8211:E
8206:H
8194:E
8189:R
8177:E
8172:R
8160:E
8153:E
8147:E
8135:E
8128:E
8123:H
8113:R
8109:R
8103:E
8098:R
8086:E
8079:E
8073:E
8067:4
8063:6
8061:E
8056:6
8054:E
8049:6
8047:E
8042:6
8040:E
8033:6
8031:E
8025:4
8021:6
8019:E
8014:C
8010:C
8006:6
8004:E
7999:C
7995:C
7991:6
7989:E
7984:C
7980:6
7978:E
7971:6
7969:E
7955:F
7946:F
7937:F
7929:F
7918:F
7908:)
7906:R
7902:R
7895:F
7886:F
7878:F
7870:4
7867:F
7861:4
7858:F
7850:4
7847:F
7839:C
7835:4
7832:F
7826:C
7822:C
7818:4
7815:F
7807:4
7805:F
7794:R
7790:R
7783:G
7774:G
7766:G
7758:2
7755:G
7747:2
7744:G
7736:C
7732:C
7728:2
7725:G
7717:2
7714:G
7698:σ
7688:R
7686:,
7684:n
7680:G
7675:n
7671:ℓ
7667:k
7663:ℓ
7661:,
7659:k
7655:G
7649:C
7647:,
7645:n
7641:G
7635:R
7633:,
7631:l
7627:R
7625:,
7623:k
7619:G
7613:R
7611:,
7609:n
7605:G
7597:H
7595:,
7593:p
7589:p
7587:,
7585:p
7579:q
7577:,
7575:p
7571:G
7565:C
7563:,
7561:n
7557:n
7555:,
7553:n
7546:q
7542:ℓ
7540:,
7537:p
7533:ℓ
7528:q
7524:k
7522:,
7520:p
7517:k
7513:G
7508:n
7504:q
7500:p
7496:q
7494:,
7492:p
7488:G
7480:C
7478:,
7476:n
7472:G
7467:n
7463:ℓ
7459:k
7455:C
7453:,
7451:ℓ
7447:C
7445:,
7443:k
7439:G
7433:C
7431:,
7429:n
7425:G
7413:H
7409:n
7405:G
7399:ℓ
7397:,
7395:k
7391:ℓ
7387:k
7383:G
7377:C
7373:n
7369:G
7363:ℓ
7361:,
7359:k
7355:ℓ
7351:k
7347:G
7341:n
7337:H
7333:n
7329:G
7321:R
7319:,
7317:n
7313:n
7311:,
7309:n
7303:q
7301:,
7299:p
7295:q
7291:p
7287:G
7281:C
7279:,
7277:n
7273:n
7271:,
7269:n
7262:q
7258:l
7256:,
7253:p
7249:ℓ
7244:q
7240:k
7238:,
7235:p
7231:k
7227:G
7221:q
7219:,
7217:p
7213:G
7205:n
7201:C
7197:n
7193:G
7188:n
7184:ℓ
7180:k
7176:C
7174:,
7172:ℓ
7168:C
7166:,
7164:k
7160:G
7154:C
7152:,
7150:n
7146:G
7135:n
7131:ℓ
7127:k
7123:ℓ
7121:,
7119:k
7115:ℓ
7111:k
7107:G
7101:C
7097:n
7093:G
7087:ℓ
7085:,
7083:k
7079:H
7075:ℓ
7071:H
7067:k
7063:G
7057:H
7053:n
7049:G
7043:n
7039:H
7035:n
7031:G
7023:R
7021:,
7019:p
7015:p
7013:,
7011:p
7005:q
7003:,
7001:p
6997:q
6993:p
6989:G
6983:C
6981:,
6979:p
6975:p
6973:,
6971:p
6964:q
6960:l
6958:,
6955:p
6951:l
6946:q
6942:k
6940:,
6938:p
6935:k
6931:G
6925:H
6923:,
6921:p
6917:p
6915:,
6913:p
6907:q
6905:,
6903:p
6899:G
6894:n
6890:q
6886:p
6882:q
6880:,
6878:p
6874:G
6866:n
6862:R
6860:,
6858:n
6854:G
6848:n
6844:C
6840:n
6836:G
6830:R
6828:,
6826:l
6822:R
6820:,
6818:k
6814:G
6808:l
6806:,
6804:k
6800:G
6794:R
6792:,
6790:n
6786:G
6778:n
6774:C
6772:,
6770:n
6766:G
6761:n
6757:ℓ
6753:k
6749:C
6747:,
6745:ℓ
6741:C
6739:,
6737:k
6733:G
6727:C
6725:,
6723:n
6719:G
6713:C
6711:,
6709:n
6705:G
6684:τ
6680:σ
6676:τ
6672:σ
6656:c
6650:g
6637:τ
6633:τ
6631:∘
6629:σ
6622:τ
6620:∘
6618:σ
6614:τ
6598:c
6592:g
6579:σ
6563:c
6557:g
6544:τ
6524:g
6500:c
6494:g
6478:K
6474:G
6470:K
6466:G
6462:G
6458:H
6454:H
6450:G
6434:g
6410:c
6404:g
6379:g
6367:σ
6351:g
6339:H
6335:H
6331:G
6324:H
6320:G
6304:g
6280:g
6255:g
6243:H
6239:G
6212:h
6188:h
6160:m
6133:m
6123:h
6118:=
6113:g
6098:H
6094:G
6068:M
6056:G
6051:H
6047:G
6043:M
6035:n
6031:n
6011:n
6007:n
5983:B
5979:A
5965:,
5960:n
5956:)
5951:B
5943:A
5939:(
5892:H
5867:O
5834:)
5831:4
5828:(
5824:O
5821:S
5796:2
5792:G
5763:2
5759:P
5754:O
5722:)
5719:9
5716:(
5712:n
5709:i
5706:p
5703:S
5678:4
5674:F
5645:2
5641:P
5636:H
5613:2
5609:P
5604:O
5574:)
5571:2
5568:(
5564:U
5561:S
5554:)
5551:3
5548:(
5544:p
5541:S
5516:4
5512:F
5483:2
5479:P
5475:)
5471:O
5463:H
5459:(
5437:2
5433:P
5429:)
5425:O
5417:O
5413:(
5384:)
5381:2
5378:(
5374:U
5371:S
5362:7
5358:E
5333:8
5329:E
5300:2
5296:P
5292:)
5288:O
5280:O
5276:(
5245:}
5241:l
5238:o
5235:v
5228:{
5224:/
5220:)
5214:(
5210:n
5207:i
5204:p
5201:S
5176:8
5172:E
5143:2
5139:P
5135:)
5131:O
5123:C
5119:(
5097:2
5093:P
5089:)
5085:O
5077:H
5073:(
5044:)
5041:2
5038:(
5034:O
5031:S
5022:6
5018:E
4993:7
4989:E
4961:O
4953:H
4930:2
4926:P
4922:)
4918:O
4910:H
4906:(
4875:)
4872:2
4869:(
4865:U
4862:S
4855:)
4849:(
4845:O
4842:S
4817:7
4813:E
4777:}
4774:I
4768:{
4764:/
4760:)
4757:8
4754:(
4750:U
4747:S
4722:7
4718:E
4689:2
4684:P
4681:O
4657:2
4653:P
4649:)
4645:O
4637:C
4633:(
4602:4
4598:F
4573:6
4569:E
4540:2
4536:P
4532:)
4528:O
4520:C
4516:(
4484:)
4481:2
4478:(
4474:O
4471:S
4464:)
4458:(
4454:O
4451:S
4426:6
4422:E
4393:2
4389:P
4385:)
4381:H
4373:C
4369:(
4347:2
4343:P
4339:)
4335:O
4327:C
4323:(
4294:)
4291:2
4288:(
4284:U
4281:S
4274:)
4271:6
4268:(
4264:U
4261:S
4236:6
4232:E
4196:}
4193:I
4187:{
4183:/
4179:)
4176:4
4173:(
4169:p
4166:S
4141:6
4137:E
4108:q
4105:+
4102:p
4097:H
4085:p
4066:)
4063:q
4060:,
4057:p
4054:(
4030:q
4027:p
4024:4
4002:)
3999:q
3996:(
3992:p
3989:S
3982:)
3979:p
3976:(
3972:p
3969:S
3946:)
3943:q
3940:+
3937:p
3934:(
3930:p
3927:S
3897:n
3892:H
3868:n
3847:)
3844:1
3841:+
3838:n
3835:(
3832:n
3810:)
3807:n
3804:(
3800:U
3777:)
3774:n
3771:(
3767:p
3764:S
3735:n
3732:2
3727:R
3703:]
3700:2
3696:/
3692:n
3689:[
3668:)
3665:1
3659:n
3656:(
3653:n
3631:)
3628:n
3625:(
3621:U
3598:)
3595:n
3592:2
3589:(
3585:O
3582:S
3553:q
3550:+
3547:p
3542:R
3530:p
3511:)
3508:q
3505:,
3502:p
3499:(
3475:q
3472:p
3450:)
3447:q
3444:(
3440:O
3437:S
3430:)
3427:p
3424:(
3420:O
3417:S
3394:)
3391:q
3388:+
3385:p
3382:(
3378:O
3375:S
3346:q
3343:+
3340:p
3335:C
3323:p
3304:)
3301:q
3298:,
3295:p
3292:(
3268:q
3265:p
3262:2
3240:)
3237:)
3234:q
3231:(
3227:U
3220:)
3217:p
3214:(
3210:U
3206:(
3202:S
3179:)
3176:q
3173:+
3170:p
3167:(
3163:U
3160:S
3130:n
3127:2
3122:C
3098:1
3092:n
3071:)
3068:1
3065:+
3062:n
3059:2
3056:(
3053:)
3050:1
3044:n
3041:(
3019:)
3016:n
3013:(
3009:p
3006:S
2983:)
2980:n
2977:2
2974:(
2970:U
2967:S
2937:n
2932:C
2908:1
2902:n
2881:2
2877:/
2873:)
2870:2
2867:+
2864:n
2861:(
2858:)
2855:1
2849:n
2846:(
2824:)
2821:n
2818:(
2814:O
2811:S
2788:)
2785:n
2782:(
2778:U
2775:S
2749:K
2744:G
2732:K
2728:G
2724:K
2720:G
2705:G
2701:G
2697:G
2693:K
2689:G
2685:K
2681:G
2674:2
2671:G
2667:4
2664:F
2660:8
2657:E
2653:7
2650:E
2646:6
2643:E
2636:n
2632:n
2628:n
2620:G
2614:K
2610:G
2606:K
2602:M
2598:M
2594:G
2590:M
2579:G
2572:G
2557:M
2549:.
2543:M
2535:.
2529:M
2518:M
2498:G
2490:K
2486:M
2482:G
2474:M
2470:g
2466:σ
2462:K
2458:G
2454:M
2429:M
2425:g
2421:σ
2417:K
2413:G
2409:M
2402:M
2398:p
2393:p
2389:s
2385:M
2381:p
2375:p
2371:s
2367:p
2363:p
2361:(
2358:p
2354:s
2349:p
2345:s
2341:K
2337:G
2333:σ
2316:K
2313:)
2306:h
2300:1
2293:h
2289:(
2283:h
2277:K
2270:h
2265:,
2262:M
2256:M
2253::
2248:p
2244:s
2230:G
2226:h
2222:K
2213:p
2208:K
2204:G
2197:K
2193:G
2189:g
2185:G
2181:G
2177:K
2169:K
2165:G
2161:K
2157:K
2153:K
2149:G
2145:M
2128:,
2122:o
2118:)
2108:G
2104:(
2078:G
2067:K
2040:p
2036:s
2029:h
2021:p
2017:s
2010:h
2007:,
2004:G
1998:G
1995::
1979:p
1975:M
1970:p
1966:s
1962:M
1958:p
1953:K
1945:M
1941:p
1936:K
1932:p
1928:p
1924:M
1920:G
1912:K
1904:M
1900:M
1896:p
1892:M
1888:M
1880:M
1876:G
1872:M
1848:m
1824:h
1812:σ
1796:g
1769:h
1745:m
1717:g
1693:h
1669:h
1645:m
1621:g
1597:h
1572:.
1567:h
1559:]
1554:m
1549:,
1544:m
1539:[
1535:,
1530:m
1522:]
1517:m
1512:,
1507:h
1502:[
1498:,
1493:h
1485:]
1480:h
1475:,
1470:h
1465:[
1437:m
1427:h
1422:=
1417:g
1386:g
1374:σ
1358:m
1346:G
1342:H
1326:h
1314:σ
1310:σ
1306:G
1290:g
1278:σ
1274:G
1267:G
1263:H
1246:.
1243:}
1240:g
1237:=
1234:)
1231:g
1228:(
1222::
1219:G
1213:g
1210:{
1207:=
1198:G
1184:H
1179:G
1174:σ
1170:G
1166:σ
1162:G
1158:σ
1151:H
1147:H
1143:G
1139:G
1127:G
1098:)
1095:1
1092:,
1089:2
1086:(
1083:L
1027:M
1023:M
1019:M
1015:M
1007:M
984:M
968:M
956:M
950:M
946:p
942:f
938:p
930:p
926:f
912:.
909:)
906:t
900:(
894:=
891:)
888:)
885:t
882:(
876:(
873:f
853:p
850:=
847:)
844:0
841:(
828:γ
824:p
816:p
812:f
808:M
804:p
800:M
781:H
769:H
765:G
758:H
754:G
729:M
724:p
720:T
709:p
705:M
701:M
697:p
693:g
689:M
687:(
616:e
609:t
602:v
242:8
240:E
234:7
232:E
226:6
224:E
218:4
216:F
210:2
208:G
190:n
187:D
180:n
177:C
170:n
167:B
160:n
157:A
128:)
126:n
118:)
116:n
108:)
106:n
98:)
96:n
88:)
86:n
78:)
76:n
68:)
66:n
25:.
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