587:. One can understand this more concretely by identifying the space of left-invariant vector fields with the tangent space at the identity, as follows: Given a left-invariant vector field, one can take its value at the identity, and given a tangent vector at the identity, one can extend it to a left-invariant vector field. This correspondence is one-to-one in both directions, so is bijective. Thus, the Lie algebra can be thought of as the tangent space at the identity and the bracket of
6603:
1854:
2659:
is simply connected cannot be omitted. For example, the Lie algebras of SO(3) and SU(2) are isomorphic, but there is no corresponding homomorphism of SO(3) into SU(2). Rather, the homomorphism goes from the simply connected group SU(2) to the non-simply connected group SO(3). If
4708:
805:
5319:, which is not simply connected. There is one irreducible representation of the Lie algebra in each dimension, but only the odd-dimensional representations of the Lie algebra come from representations of the group. (This observation is related to the distinction between
8456:
3025:
6721:
3436:
1975:
5987:
7891:
4936:
6472:
9535:
8149:
8718:
6929:
3154:
8595:
8297:
1171:
8854:
6804:
3811:
7257:
6428:
5540:
6292:
1734:
5175:
940:
8020:
1742:
2486:
8944:
5590:
9822:
9183:
4458:
4419:
3748:
3645:
2403:
698:
3694:
8320:
1099:
5876:
5768:
5840:
1244:
8204:
7495:
6185:
7943:
7541:
2269:
2087:
8767:
1611:
8502:
6089:
5098:
427:
3266:
2888:
7088:
3231:
6609:
3347:
7002:
5697:
1383:
262:
4794:
3062:
9268:
9035:
6851:
4024:
3992:
2617:
2313:
1459:
1427:
658:
577:
497:
7635:
4120:
3443:
2162:
1021:
5038:
3485:
3342:
5804:
5638:
2221:
5916:
7781:
6598:{\displaystyle {\mathfrak {z}}_{\mathfrak {g}}(A)=\{X\in {\mathfrak {g}}\mid \operatorname {ad} (a)X=0{\text{ or }}\operatorname {Ad} (a)X=0{\text{ for all }}a{\text{ in }}A\}}
5415:
3896:
3839:
3514:
7711:
7458:
7366:
7313:
7167:
4802:
4307:
3580:
3543:
1661:
95:
66:
9846:
9296:
8975:
7666:
7603:
7572:
7390:
7337:
7284:
6460:
5904:
5721:
5618:
5469:
5305:
5277:
3924:
3086:
2649:
2585:
7005:
2880:
from (finite dimensional) Lie algebras to Lie groups (which is necessarily unique up to canonical isomorphism). In other words there is a natural isomorphism of bifunctors
2819:
304:
7134:
1893:
349:
5230:
4257:
1888:
1519:
6320:
4056:
2551:
2522:
2017:
1318:
1286:
855:
7748:
7030:
4442:
3174:
2878:
5445:
2748:
623:
7050:
5205:
4731:
4212:
4185:
4150:
3960:
458:
9325:
9216:
9109:
9068:
8043:
6019:
3296:
2848:
8618:
2040:
5253:
2768:
2713:
6856:
537:
3095:
625:
can be computed by extending them to left-invariant vector fields, taking the bracket of the vector fields, and then evaluating the result at the identity.
9430:
8507:
8209:
1111:
7094:
8772:
6733:
3756:
7186:
6329:
5474:
6193:
1666:
5113:
1849:{\displaystyle \operatorname {Lie} (G_{1}\times \cdots \times G_{r})=\operatorname {Lie} (G_{1})\oplus \cdots \oplus \operatorname {Lie} (G_{r}).}
870:
10875:
7948:
10066:
2435:
4953:
are sufficiently small. This argument is only local, since the exponential map is only invertible in a small neighborhood of the identity in
10870:
39:
to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is
8879:
5545:
5327:
is simply connected with Lie algebra isomorphic to that of SO(3), so every representation of the Lie algebra of SO(3) does give rise to a
4703:{\displaystyle f\left(e^{X}e^{Y}\right)=f\left(e^{Z}\right)=e^{\phi (Z)}=e^{\phi (X)+\phi (Y)+{\frac {1}{2}}+{\frac {1}{12}}]+\cdots },}
10157:
10181:
9762:
9122:
105:
respectively) which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, for
4315:
3702:
9356:
4734:
3939:
2681:
3585:
2271:
defines a local homeomorphism from a neighborhood of the zero vector to the neighborhood of the identity element. For example, if
5593:
800:{\displaystyle \operatorname {Lie} (G)=\left\{X\in M(n;\mathbb {C} )\mid e^{tX}\in G{\text{ for all }}t\in \mathbb {R} \right\}.}
8451:{\displaystyle =\left\{A\in U(2n)\mid A^{\mathrm {T} }JA=J\right\},\,J={\begin{bmatrix}0&I_{n}\\-I_{n}&0\end{bmatrix}}}
3938:
One approach to proving the second part of the Lie group-Lie algebra correspondence (the homomorphisms theorem) is to use the
10246:
9977:
3657:
2322:
3310:, which says any finite-dimensional Lie algebra (over a field of any characteristic) is a Lie subalgebra of the Lie algebra
1039:
10472:
5845:
5737:
9936:
5809:
1188:
10525:
10053:
8156:
7470:
6130:
7898:
7503:
2230:
2048:
10809:
9927:
8725:
1564:
10009:
8463:
3020:{\displaystyle \mathrm {Hom} _{CLGrp}(\Gamma ({\mathfrak {g}}),H)\cong \mathrm {Hom} _{LAlg}({\mathfrak {g}},Lie(H)).}
9948:
4993:
10025:
6716:{\displaystyle Z_{G}(A)=\{g\in G\mid \operatorname {Ad} (g)a=0{\text{ or }}ga=ag{\text{ for all }}a{\text{ in }}A\}}
6047:
5050:
3431:{\displaystyle {\mathfrak {g}}\subset {\mathfrak {gl}}_{n}(\mathbb {R} )=\operatorname {Lie} (GL_{n}(\mathbb {R} ))}
354:
10574:
9969:
9186:
9079:
7672:
It is important to emphasize that the equivalence of the preceding conditions holds only under the assumption that
3236:
7055:
7004:
is a surjective group homomorphism. The kernel of it is a discrete group (since the dimension is zero) called the
5235:
The homomorphisms theorem (mentioned above as part of the Lie group-Lie algebra correspondence) then says that if
3186:
10557:
10166:
7178:
157:
6971:
5643:
3902:
its image. The preceding can be summarized to saying that there is a canonical bijective correspondence between
1323:
192:
10918:
9901:
5328:
140:
4744:
3034:
2854:
of connected (real) Lie groups to the category of finite-dimensional (real) Lie-algebras. This functor has a
10769:
10176:
10019:
9965:
9225:
9008:
6824:
5324:
3997:
3965:
2590:
2286:
1432:
1400:
631:
550:
470:
7608:
4064:
2096:
955:
10754:
10477:
10251:
9271:
5982:{\displaystyle 0\to Z(G)\to G\xrightarrow {\operatorname {Ad} } \operatorname {Int} ({\mathfrak {g}})\to 0}
2043:
1032:
504:
7886:{\displaystyle =\left\{A\in M_{n+1}(\mathbb {C} )\mid {\overline {A}}^{\mathrm {T} }A=I,\det(A)=1\right\}}
5002:
3449:
3313:
10799:
10014:
5773:
5623:
4931:{\displaystyle f\left(e^{X}e^{Y}\right)=e^{\phi (X)}e^{\phi (Y)}=f\left(e^{X}\right)f\left(e^{Y}\right).}
2183:
10804:
10774:
10482:
10438:
10419:
10186:
10130:
9071:
5362:
5181:
3860:
3816:
3490:
3442:
be the closed (without taking the closure one can get pathological dense example as in the case of the
430:
7687:
7434:
7342:
7289:
7143:
4262:
3556:
3519:
1620:
164:
71:
42:
10341:
10206:
9827:
9277:
8956:
7647:
7584:
7553:
7371:
7318:
7265:
6441:
5885:
5702:
5599:
5450:
5286:
5258:
3905:
3546:
3269:
3177:
3067:
2668:
are both simply connected and have isomorphic Lie algebras, the above result allows one to show that
2630:
2566:
1174:
110:
3813:
and exp for it is the identity, this homomorphism is the differential of the Lie group homomorphism
1970:{\displaystyle \operatorname {Lie} (H\cap H')=\operatorname {Lie} (H)\cap \operatorname {Lie} (H').}
10726:
10591:
10283:
10125:
9341:
9219:
5320:
2773:
1614:
1534:
946:
811:
2413:
The correspondence between Lie groups and Lie algebras includes the following three main results.
270:
10423:
10393:
10317:
10307:
10263:
10093:
10046:
7103:
2423:
1470:
685:
309:
4225:
10764:
10383:
10278:
10191:
10098:
8950:
5340:
3850:
2851:
2418:
1484:
861:
106:
6299:
4029:
2527:
2495:
1990:
1291:
1259:
828:
10923:
10413:
10408:
8144:{\displaystyle =\left\{A\in M_{2n+1}(\mathbb {R} )\mid A^{\mathrm {T} }A=I,\det(A)=1\right\}}
7720:
7638:
7015:
4427:
3159:
2863:
2691:
the correspondence can be summarised as follows: First, the operation of associating to each
98:
5420:
2718:
598:
10744:
10682:
10530:
10234:
10224:
10196:
10171:
10081:
8713:{\displaystyle =\left\{A\in M_{2n}(\mathbb {R} )\mid A^{\mathrm {T} }A=I,\det(A)=1\right\}}
7035:
5316:
5210:
5187:
4716:
4190:
4163:
4128:
3945:
2280:
1550:
436:
9301:
9192:
9085:
9044:
5995:
1862:
8:
10928:
10882:
10564:
10442:
10427:
10356:
10115:
9336:
6924:{\displaystyle \operatorname {Lie} (G/H)=\operatorname {Lie} (G)/\operatorname {Lie} (H)}
3275:
2827:
10855:
3149:{\displaystyle \epsilon \colon {\mathfrak {g}}\rightarrow Lie(\Gamma ({\mathfrak {g}}))}
2022:
10933:
10824:
10779:
10676:
10547:
10351:
10039:
9351:
8978:
5238:
2753:
2698:
10361:
510:
10759:
10739:
10734:
10641:
10552:
10366:
10346:
10201:
10140:
9983:
9973:
9944:
9923:
9897:
9530:{\displaystyle \operatorname {Lie} (f^{-1}(H'))=(df)^{-1}(\operatorname {Lie} (H')).}
7412:
7400:
7098:
6940:
3942:, as in Section 5.7 of Hall's book. Specifically, given the Lie algebra homomorphism
121:
35:
or vice versa, and study the conditions for such a relationship. Lie groups that are
8590:{\displaystyle =\left\{X\in M_{2n}(\mathbb {C} )\mid X^{\mathrm {T} }J+JX=0\right\}}
8292:{\displaystyle =\left\{X\in M_{2n+1}(\mathbb {C} )\mid X^{\mathrm {T} }+X=0\right\}}
1166:{\displaystyle \operatorname {Lie} (\operatorname {im} (f))=\operatorname {im} (df)}
10897:
10691:
10646:
10569:
10540:
10398:
10331:
10326:
10321:
10311:
10103:
10086:
10002:
9915:
9889:
7759:
6041:
3307:
1546:
136:
125:
8849:{\displaystyle =\left\{X\in M_{2n}(\mathbb {C} )\mid X^{\mathrm {T} }+X=0\right\}}
5878:
is in general not a closed subgroup; only an immersed subgroup.) It is called the
10840:
10749:
10579:
10535:
10301:
6799:{\displaystyle \operatorname {Lie} (Z_{G}(A))={\mathfrak {z}}_{\mathfrak {g}}(A)}
3806:{\displaystyle \operatorname {Lie} (\mathbb {R} )=T_{0}\mathbb {R} =\mathbb {R} }
2858:
2688:
660:
as the Lie algebra of primitive elements of the Hopf algebra of distributions on
7421:
be a connected Lie group with finite center. Then the following are equivalent.
5542:, called the adjoint representation. The corresponding Lie algebra homomorphism
10706:
10631:
10601:
10499:
10492:
10432:
10403:
10273:
10268:
10229:
8990:
7252:{\displaystyle 1\to \pi _{1}(G)\to {\widetilde {G}}{\overset {p}{\to }}G\to 1.}
6423:{\displaystyle T_{x}(G\cdot x)=\operatorname {im} (d\rho (x):T_{e}G\to T_{x}X)}
9919:
9893:
5535:{\displaystyle \operatorname {Ad} :G\to GL({\mathfrak {g}}),\,g\mapsto dc_{g}}
3064:
is the (up to isomorphism unique) simply-connected Lie group with Lie algebra
10912:
10892:
10716:
10711:
10696:
10686:
10636:
10613:
10487:
10447:
10388:
10336:
10135:
9987:
7396:
6287:{\displaystyle \operatorname {Lie} (G_{x})=\ker(d\rho (x):T_{e}G\to T_{x}X).}
5879:
4976:
from a local homomorphism to a global one. The extension is done by defining
1729:{\displaystyle dp_{i}:\operatorname {Lie} (G)\to \operatorname {Lie} (G_{i})}
461:
116:
In this article, a Lie group refers to a real Lie group. For the complex and
8985:. (Roughly, this is a consequence of the fact that any differential form on
5315:
be simply connected is essential. Consider, for example, the rotation group
5170:{\displaystyle d\pi :{\mathfrak {g}}\to {\mathfrak {gl}}_{n}(\mathbb {C} ),}
3582:
is a Lie group and that the covering map is a Lie group homomorphism. Since
935:{\displaystyle df=df_{e}:\operatorname {Lie} (G)\to \operatorname {Lie} (H)}
10819:
10814:
10656:
10623:
10596:
10504:
10145:
9346:
9112:
7547:
3344:
of square matrices. The proof goes as follows: by Ado's theorem, we assume
2176:
is a neighborhood of the identity element in a connected topological group
810:
For example, one can use the criterion to establish the correspondence for
102:
9545:
8015:{\displaystyle =\{X\in M_{n+1}(\mathbb {C} )\mid \operatorname {tr} X=0\}}
7368:
by discrete central subgroups and connected Lie groups having Lie algebra
10662:
10651:
10608:
10509:
10110:
9957:
9914:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
9912:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
7765:
2619:, then there is a unique connected Lie subgroup (not necessarily closed)
2492:
is simply connected, then there exists a (unique) Lie group homomorphism
36:
32:
20:
2481:{\displaystyle \phi :\operatorname {Lie} (G)\to \operatorname {Lie} (H)}
10887:
10845:
10671:
10584:
10216:
10120:
10031:
2422:: Every finite-dimensional real Lie algebra is the Lie algebra of some
1253:
9078:
with support at the identity element with the multiplication given by
7750:. The last three conditions above are purely Lie algebraic in nature.
10701:
10666:
10371:
10258:
5724:
5041:
28:
8939:{\displaystyle H^{k}({\mathfrak {g}};\mathbb {R} )=H_{\text{dR}}(G)}
5948:
5585:{\displaystyle {\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})}
4160:
is a local homomorphism. Thus, given two elements near the identity
4156:, which has an inverse defined near the identity. We now argue that
10865:
10860:
10850:
10241:
10062:
4444:
indicating other terms expressed as repeated commutators involving
132:
4996:
of a Lie group and representations of the associated Lie algebra.
4984:
to show that the definition is independent of the choice of path.
4058:
locally (i.e., in a neighborhood of the identity) by the formula
156:
There are various ways one can understand the construction of the
6949:
be a connected Lie group. Since the Lie algebra of the center of
4992:
A special case of Lie correspondence is a correspondence between
2822:
6726:
be the Lie algebra centralizer and the Lie group centralizer of
10457:
9817:{\displaystyle \exp({\mathfrak {g}})^{n}=\exp({\mathfrak {g}})}
9178:{\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)=P(A(G))}
949:(brackets go to brackets), which has the following properties:
579:
is a Lie subalgebra of the Lie algebra of all vector fields on
4414:{\displaystyle Z=X+Y+{\frac {1}{2}}+{\frac {1}{12}}]+\cdots ,}
3743:{\displaystyle \mathbb {R} \to {\mathfrak {g}},\,t\mapsto tX.}
3306:
Perhaps the most elegant proof of the first result above uses
2655:
In the second part of the correspondence, the assumption that
499:
be the set of all left-translation-invariant vector fields on
8722:
8609:
8460:
8311:
8153:
8034:
7895:
7772:
4957:
and since the Baker–Campbell–Hausdorff formula only holds if
4259:. According to the Baker–Campbell–Hausdorff formula, we have
7339:, there is a one-to-one correspondence between quotients of
3640:{\displaystyle T_{e}{\widetilde {G}}=T_{e}G={\mathfrak {g}}}
2275:
is the Lie group of invertible real square matrices of size
2227:, since the former is an open (hence closed) subgroup. Now,
174:
is said to be invariant under left translations if, for any
7713:
is also compact. Clearly, this conclusion does not hold if
503:. It is a real vector space. Moreover, it is closed under
3156:
of the adjunction are isomorphisms, which corresponds to
4980:
along a path and then using the simple connectedness of
3180:(part of the second statement above). The corresponding
2398:{\textstyle \exp(X)=e^{X}=\sum _{0}^{\infty }{X^{j}/j!}}
109:
Lie groups, the Lie group-Lie algebra correspondence is
5255:
is the simply connected Lie group whose Lie algebra is
3689:{\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)}
9941:
Differential geometry, Lie groups and symmetric spaces
8400:
6961:
is abelian if and only if its Lie algebra is abelian.
6322:
is locally closed, then the orbit is a submanifold of
5699:, which in particular implies that the Lie bracket of
2325:
2186:
1094:{\displaystyle \operatorname {Lie} (\ker(f))=\ker(df)}
9830:
9765:
9433:
9304:
9280:
9228:
9195:
9125:
9088:
9047:
9011:
8959:
8882:
8775:
8728:
8621:
8510:
8466:
8323:
8212:
8159:
8046:
7951:
7901:
7784:
7723:
7690:
7650:
7611:
7587:
7556:
7506:
7473:
7437:
7374:
7345:
7321:
7292:
7268:
7189:
7146:
7140:
is a central subgroup of a simply connected covering
7106:
7058:
7038:
7018:
6974:
6859:
6827:
6736:
6612:
6475:
6444:
6332:
6302:
6196:
6133:
6050:
5998:
5919:
5888:
5871:{\displaystyle \operatorname {Int} ({\mathfrak {g}})}
5848:
5812:
5776:
5763:{\displaystyle \operatorname {Int} ({\mathfrak {g}})}
5740:
5705:
5646:
5626:
5602:
5548:
5477:
5453:
5423:
5365:
5289:
5261:
5241:
5213:
5190:
5116:
5053:
5005:
4805:
4747:
4719:
4461:
4430:
4318:
4265:
4228:
4193:
4166:
4131:
4067:
4032:
4000:
3968:
3948:
3933:
3908:
3863:
3819:
3759:
3705:
3660:
3588:
3559:
3522:
3493:
3452:
3350:
3316:
3278:
3239:
3189:
3162:
3098:
3070:
3037:
2891:
2866:
2830:
2776:
2756:
2721:
2701:
2633:
2593:
2569:
2530:
2498:
2438:
2289:
2233:
2099:
2051:
2025:
1993:
1896:
1865:
1745:
1669:
1623:
1567:
1487:
1435:
1403:
1326:
1294:
1262:
1191:
1114:
1042:
958:
873:
831:
701:
634:
601:
553:
513:
473:
439:
357:
312:
273:
195:
74:
45:
9968:. Vol. 218 (Second ed.). New York London:
9738:, Ch. III, § 3, no. 16, Corollary to Proposition 55.
5835:{\displaystyle \operatorname {ad} ({\mathfrak {g}})}
5323:
in quantum mechanics.) On the other hand, the group
1239:{\displaystyle G/\ker(f)\to \operatorname {im} (f).}
664:
with support at the identity element; for this, see
163:. One approach uses left-invariant vector fields. A
8199:{\displaystyle {\mathfrak {so}}(2n+1,\mathbb {C} )}
7490:{\displaystyle \operatorname {Int} {\mathfrak {g}}}
6180:{\displaystyle \rho (x):G\to X,\,g\mapsto g\cdot x}
2315:is the Lie algebra of real square matrices of size
9840:
9816:
9529:
9319:
9290:
9262:
9210:
9177:
9103:
9062:
9029:
8969:
8938:
8848:
8761:
8712:
8589:
8496:
8450:
8291:
8198:
8143:
8014:
7938:{\displaystyle {\mathfrak {sl}}(n+1,\mathbb {C} )}
7937:
7885:
7742:
7705:
7660:
7629:
7597:
7566:
7536:{\displaystyle G\hookrightarrow O(n,\mathbb {R} )}
7535:
7489:
7452:
7384:
7360:
7331:
7307:
7278:
7251:
7161:
7128:
7082:
7044:
7024:
6996:
6923:
6845:
6798:
6715:
6597:
6454:
6422:
6314:
6286:
6179:
6083:
6013:
5981:
5898:
5870:
5834:
5798:
5762:
5715:
5691:
5632:
5612:
5584:
5534:
5463:
5439:
5409:
5299:
5271:
5247:
5224:
5199:
5169:
5092:
5032:
4930:
4788:
4725:
4702:
4436:
4413:
4301:
4251:
4206:
4179:
4144:
4114:
4050:
4018:
3986:
3954:
3918:
3890:
3833:
3805:
3742:
3688:
3639:
3574:
3537:
3508:
3479:
3430:
3336:
3290:
3260:
3225:
3168:
3148:
3080:
3056:
3019:
2872:
2842:
2813:
2762:
2742:
2707:
2643:
2611:
2579:
2545:
2516:
2480:
2397:
2307:
2264:{\displaystyle \exp :\operatorname {Lie} (G)\to G}
2263:
2215:
2156:
2082:{\displaystyle \exp :\operatorname {Lie} (G)\to G}
2081:
2034:
2011:
1969:
1882:
1848:
1728:
1655:
1605:
1513:
1453:
1421:
1377:
1312:
1280:
1238:
1165:
1093:
1015:
934:
849:
799:
652:
617:
571:
531:
491:
452:
421:
343:
298:
256:
139:; in particular, they have at most countably many
89:
60:
9369:
8762:{\displaystyle {\mathfrak {so}}(2n,\mathbb {C} )}
1606:{\displaystyle G=G_{1}\times \cdots \times G_{r}}
151:
10910:
8687:
8497:{\displaystyle {\mathfrak {sp}}(n,\mathbb {C} )}
8118:
7860:
6051:
5734:By Lie's third theorem, there exists a subgroup
5339:An example of a Lie group representation is the
1987:is a Lie group, then any Lie group homomorphism
864:, then its differential at the identity element
10003:Notes for Math 261A Lie groups and Lie algebras
9883:
5910:is connected, it fits into the exact sequence:
3301:
814:(cf. the table in "compact Lie groups" below.)
6084:{\displaystyle \det(\operatorname {Ad} (g))=1}
5093:{\displaystyle \pi :G\to GL_{n}(\mathbb {C} )}
422:{\displaystyle (dL_{g})_{h}:T_{h}G\to T_{gh}G}
10047:
9544:This requirement cannot be omitted; see also
9041:may be alternatively defined as follows. Let
7641:and has zero or purely imaginary eigenvalues.
6029:is discrete, then Ad here is a covering map.
5334:
4945:has the homomorphism property, at least when
3261:{\displaystyle {\widetilde {H}}\rightarrow H}
2770:of Lie groups the corresponding differential
135:(in particular Lie groups) are assumed to be
8009:
7955:
7083:{\displaystyle {\mathfrak {g}}/\Gamma \to G}
6710:
6635:
6592:
6504:
5180:is then a Lie algebra homomorphism called a
5103:is called a representation of the Lie group
4987:
4972:The next stage in the argument is to extend
3226:{\displaystyle \Gamma (Lie(H))\rightarrow H}
1477:is said to be an immersed (Lie) subgroup of
9005:be a Lie group. The associated Lie algebra
7644:There exists an invariant inner product on
5447:is then an automorphism of the Lie algebra
4969:is simply connected has not yet been used.
4741:, we see that this last expression becomes
3696:gives rise to the Lie algebra homomorphism
2019:is uniquely determined by its differential
16:Correspondence between topics in Lie theory
10054:
10040:
6997:{\displaystyle \exp :{\mathfrak {g}}\to G}
5692:{\displaystyle \operatorname {ad} (X)(Y)=}
3926:and the set of one-parameter subgroups of
3849:. This Lie group homomorphism, called the
1378:{\displaystyle d(g\circ f)=(dg)\circ (df)}
665:
257:{\displaystyle (dL_{g})_{h}(X_{h})=X_{gh}}
8907:
8807:
8752:
8653:
8542:
8487:
8388:
8250:
8189:
8084:
7984:
7928:
7819:
7676:has finite center. Thus, for example, if
7526:
6161:
5512:
5157:
5083:
5023:
4733:is a Lie algebra homomorphism. Using the
3821:
3799:
3791:
3770:
3724:
3707:
3470:
3418:
3382:
2821:at the neutral element, is a (covariant)
785:
744:
77:
48:
10061:
10026:Formal Lie theory in characteristic zero
9935:
9874:
9857:
9747:
9735:
9723:
9568:
9556:
9387:
9357:Distribution on a linear algebraic group
8996:
4789:{\displaystyle e^{\phi (X)}e^{\phi (Y)}}
3057:{\displaystyle \Gamma ({\mathfrak {g}})}
2676:are isomorphic. One method to construct
9877:Groupes et Algèbres de Lie (Chapitre 3)
9546:https://math.stackexchange.com/q/329753
9263:{\displaystyle U({\mathfrak {g}})=A(G)}
9030:{\displaystyle \operatorname {Lie} (G)}
6846:{\displaystyle \operatorname {Lie} (H)}
4019:{\displaystyle \operatorname {Lie} (H)}
3987:{\displaystyle \operatorname {Lie} (G)}
2612:{\displaystyle \operatorname {Lie} (G)}
2308:{\displaystyle \operatorname {Lie} (G)}
1890:are Lie subgroups of a Lie group, then
1454:{\displaystyle \operatorname {Lie} (G)}
1422:{\displaystyle \operatorname {Lie} (H)}
1181:induces the isomorphism of Lie groups:
653:{\displaystyle \operatorname {Lie} (G)}
572:{\displaystyle \operatorname {Lie} (G)}
492:{\displaystyle \operatorname {Lie} (G)}
10911:
9750:, Ch. III, § 1, no. 7, Proposition 14.
7630:{\displaystyle \operatorname {ad} (X)}
4115:{\displaystyle f(e^{X})=e^{\phi (X)},}
2157:{\displaystyle f(\exp(X))=\exp(df(X))}
1549:with the structure group its kernel. (
1016:{\displaystyle \exp(df(X))=f(\exp(X))}
10035:
10007:
9559:, Ch. III, § 3, no. 8, Proposition 28
9222:, there is the canonical isomorphism
7431:(Weyl) The simply connected covering
7406:
6968:is abelian, then the exponential map
6934:
2488:is a Lie algebra homomorphism and if
2408:
628:There is also another incarnation of
9909:
9884:Duistermaat, J.J.; Kolk, A. (2000),
9711:
9700:
9688:
9676:
9664:
9652:
9640:
9628:
9616:
9604:
9592:
9580:
9399:
7032:. By the first isomorphism theorem,
6953:is the center of the Lie algebra of
6106:be a Lie group acting on a manifold
5471:. This way, we get a representation
5033:{\displaystyle GL_{n}(\mathbb {C} )}
3480:{\displaystyle GL_{n}(\mathbb {R} )}
3337:{\displaystyle {\mathfrak {gl}}_{n}}
1663:projections. Then the differentials
1393:is a closed subgroup of a Lie group
671:
25:Lie group–Lie algebra correspondence
9956:
9879:, Éléments de Mathématique, Hermann
9833:
9806:
9777:
9375:
9283:
9237:
9128:
8962:
8898:
8734:
8731:
8472:
8469:
8165:
8162:
7907:
7904:
7653:
7590:
7559:
7482:
7377:
7324:
7271:
7061:
6983:
6781:
6774:
6515:
6486:
6479:
6447:
5965:
5891:
5860:
5824:
5799:{\displaystyle GL({\mathfrak {g}})}
5788:
5752:
5708:
5633:{\displaystyle \operatorname {ad} }
5605:
5574:
5564:
5561:
5551:
5501:
5456:
5292:
5264:
5142:
5139:
5128:
3911:
3857:, is precisely the exponential map
3716:
3663:
3632:
3500:
3367:
3364:
3353:
3323:
3320:
3135:
3107:
3073:
3046:
2985:
2933:
2636:
2572:
2216:{\textstyle \bigcup _{n>0}U^{n}}
1736:give the canonical identification:
1556:
13:
10010:"Lie algebra of an analytic group"
8823:
8669:
8558:
8362:
8266:
8100:
7842:
7286:and a simply connected Lie group
7262:Equivalently, given a Lie algebra
7071:
7019:
6813:is a closed connected subgroup of
5321:integer spin and half-integer spin
4994:finite-dimensional representations
4222:small), we consider their product
3934:Proof of the homomorphisms theorem
3272:; its surjectivity corresponds to
3190:
3163:
3127:
3038:
2961:
2958:
2955:
2925:
2900:
2897:
2894:
2867:
2367:
1320:are Lie group homomorphisms, then
14:
10945:
9996:
5410:{\displaystyle c_{g}(h)=ghg^{-1}}
3891:{\displaystyle t\mapsto \exp(tX)}
3834:{\displaystyle \mathbb {R} \to H}
3509:{\displaystyle e^{\mathfrak {g}}}
2557:The subgroups–subalgebras theorem
583:and is called the Lie algebra of
539:is left-translation-invariant if
9962:Introduction to Smooth Manifolds
8949:where the left-hand side is the
7706:{\displaystyle {\widetilde {G}}}
7453:{\displaystyle {\widetilde {G}}}
7361:{\displaystyle {\widetilde {G}}}
7308:{\displaystyle {\widetilde {G}}}
7162:{\displaystyle {\widetilde {G}}}
4735:Baker–Campbell–Hausdorff formula
4302:{\displaystyle e^{X}e^{Y}=e^{Z}}
3940:Baker–Campbell–Hausdorff formula
3575:{\displaystyle {\widetilde {G}}}
3538:{\displaystyle {\widetilde {G}}}
2682:Baker–Campbell–Hausdorff formula
1656:{\displaystyle p_{i}:G\to G_{i}}
817:
684:), and thus a Lie group, by the
90:{\displaystyle \mathbb {T} ^{n}}
61:{\displaystyle \mathbb {R} ^{n}}
9851:
9841:{\displaystyle {\mathfrak {g}}}
9753:
9741:
9729:
9717:
9705:
9694:
9682:
9670:
9658:
9646:
9634:
9622:
9610:
9598:
9390:, Ch. II, § 2, Proposition 2.7.
9291:{\displaystyle {\mathfrak {g}}}
8977:and the right-hand side is the
8970:{\displaystyle {\mathfrak {g}}}
7661:{\displaystyle {\mathfrak {g}}}
7598:{\displaystyle {\mathfrak {g}}}
7567:{\displaystyle {\mathfrak {g}}}
7385:{\displaystyle {\mathfrak {g}}}
7332:{\displaystyle {\mathfrak {g}}}
7279:{\displaystyle {\mathfrak {g}}}
6853:is an ideal and in such a case
6455:{\displaystyle {\mathfrak {g}}}
6036:be a connected Lie group. Then
5899:{\displaystyle {\mathfrak {g}}}
5716:{\displaystyle {\mathfrak {g}}}
5613:{\displaystyle {\mathfrak {g}}}
5464:{\displaystyle {\mathfrak {g}}}
5307:comes from a representation of
5300:{\displaystyle {\mathfrak {g}}}
5272:{\displaystyle {\mathfrak {g}}}
5044:and any Lie group homomorphism
4965:are small. The assumption that
4737:again, this time for the group
3919:{\displaystyle {\mathfrak {g}}}
3444:irrational winding of the torus
3081:{\displaystyle {\mathfrak {g}}}
2644:{\displaystyle {\mathfrak {h}}}
2580:{\displaystyle {\mathfrak {h}}}
10094:Differentiable/Smooth manifold
9811:
9801:
9783:
9772:
9586:
9574:
9562:
9550:
9538:
9521:
9518:
9507:
9498:
9486:
9476:
9470:
9467:
9456:
9440:
9405:
9393:
9381:
9314:
9308:
9257:
9251:
9242:
9232:
9205:
9199:
9172:
9169:
9163:
9157:
9148:
9142:
9098:
9092:
9057:
9051:
9024:
9018:
8933:
8927:
8911:
8893:
8811:
8803:
8756:
8739:
8696:
8690:
8657:
8649:
8546:
8538:
8491:
8477:
8350:
8341:
8254:
8246:
8193:
8170:
8127:
8121:
8088:
8080:
7988:
7980:
7932:
7912:
7869:
7863:
7823:
7815:
7717:has infinite center, e.g., if
7624:
7618:
7530:
7516:
7510:
7243:
7232:
7215:
7212:
7206:
7193:
7123:
7117:
7074:
6988:
6918:
6912:
6898:
6892:
6880:
6866:
6840:
6834:
6793:
6787:
6765:
6762:
6756:
6743:
6662:
6656:
6629:
6623:
6564:
6558:
6535:
6529:
6498:
6492:
6417:
6401:
6382:
6376:
6367:
6355:
6343:
6278:
6262:
6243:
6237:
6228:
6216:
6203:
6165:
6152:
6143:
6137:
6072:
6069:
6063:
6054:
6008:
6002:
5973:
5970:
5960:
5938:
5935:
5929:
5923:
5865:
5855:
5829:
5819:
5793:
5783:
5757:
5747:
5686:
5674:
5668:
5662:
5659:
5653:
5579:
5569:
5556:
5516:
5506:
5496:
5487:
5382:
5376:
5161:
5153:
5133:
5087:
5079:
5063:
5027:
5019:
4875:
4869:
4856:
4850:
4781:
4775:
4762:
4756:
4686:
4683:
4680:
4674:
4665:
4659:
4653:
4647:
4641:
4635:
4619:
4616:
4610:
4601:
4595:
4589:
4573:
4567:
4558:
4552:
4536:
4530:
4399:
4396:
4384:
4375:
4359:
4347:
4104:
4098:
4084:
4071:
4042:
4013:
4007:
3981:
3975:
3885:
3876:
3867:
3825:
3774:
3766:
3728:
3711:
3683:
3677:
3553:; it is not hard to show that
3474:
3466:
3425:
3422:
3414:
3398:
3386:
3378:
3252:
3217:
3214:
3211:
3205:
3193:
3143:
3140:
3130:
3124:
3112:
3051:
3041:
3011:
3008:
3002:
2980:
2947:
2938:
2928:
2922:
2792:
2786:
2737:
2731:
2606:
2600:
2508:
2475:
2469:
2460:
2457:
2451:
2338:
2332:
2302:
2296:
2255:
2252:
2246:
2168:is connected, this determines
2151:
2148:
2142:
2133:
2121:
2118:
2112:
2103:
2073:
2070:
2064:
2003:
1961:
1950:
1938:
1932:
1920:
1903:
1840:
1827:
1809:
1796:
1784:
1752:
1723:
1710:
1701:
1698:
1692:
1640:
1508:
1502:
1448:
1442:
1416:
1410:
1372:
1363:
1357:
1348:
1342:
1330:
1304:
1272:
1230:
1224:
1215:
1212:
1206:
1160:
1151:
1139:
1136:
1130:
1121:
1088:
1079:
1067:
1064:
1058:
1049:
1010:
1007:
1001:
992:
983:
980:
974:
965:
929:
923:
914:
911:
905:
841:
748:
734:
714:
708:
647:
641:
566:
560:
526:
514:
486:
480:
400:
375:
358:
329:
323:
290:
235:
222:
213:
196:
152:The Lie algebra of a Lie group
1:
10028:, a blog post by Akhil Mathew
9966:Graduate Texts in Mathematics
9867:
9571:, Ch. III, § 1, Proposition 5
8873:is a compact Lie group, then
3233:is the canonical projection
2814:{\displaystyle Lie(f)=df_{e}}
1983:be a connected Lie group. If
680:is a closed subgroup of GL(n;
9362:
9272:universal enveloping algebra
8993:by the averaging argument.)
7835:
3647:, this completes the proof.
3302:Proof of Lie's third theorem
299:{\displaystyle L_{g}:G\to G}
7:
10800:Classification of manifolds
10015:Encyclopedia of Mathematics
9330:
7129:{\displaystyle \pi _{1}(G)}
5355:defines an automorphism of
5207:is often simply denoted by
4152:is the exponential map for
3841:for some immersed subgroup
3753:By Lie's third theorem, as
2750:, and to each homomorphism
1521:is an immersed subgroup of
344:{\displaystyle L_{g}(x)=gx}
27:allows one to correspond a
10:
10950:
9888:, Universitext, Springer,
7762:of associated Lie algebra
7500:There exists an embedding
7410:
6938:
6119:the stabilizer of a point
5335:The adjoint representation
5182:Lie algebra representation
4252:{\displaystyle e^{X}e^{Y}}
3298:being a faithful functor.
3088:. The associated natural
2687:For readers familiar with
2424:simply connected Lie group
2042:. Precisely, there is the
688:. Then the Lie algebra of
158:Lie algebra of a Lie group
10876:over commutative algebras
10833:
10792:
10725:
10622:
10518:
10465:
10456:
10292:
10215:
10154:
10074:
9920:10.1007/978-3-319-13467-3
9894:10.1007/978-3-642-56936-4
7136:of a connected Lie group
6821:is normal if and only if
4999:The general linear group
4988:Lie group representations
3547:simply connected covering
3438:is a Lie subalgebra. Let
3270:simply connected covering
2430:The homomorphisms theorem
2172:uniquely. In general, if
1514:{\displaystyle G/\ker(f)}
1175:first isomorphism theorem
146:
10592:Riemann curvature tensor
9759:It's surjective because
9423:is a closed subgroup of
7052:induces the isomorphism
6315:{\displaystyle G\cdot x}
4796:, and therefore we have
4051:{\displaystyle f:G\to H}
2546:{\displaystyle \phi =df}
2517:{\displaystyle f:G\to H}
2012:{\displaystyle f:G\to H}
1313:{\displaystyle g:H\to K}
1281:{\displaystyle f:G\to H}
947:Lie algebra homomorphism
850:{\displaystyle f:G\to H}
812:classical compact groups
686:closed subgroups theorem
9910:Hall, Brian C. (2015),
7743:{\displaystyle G=S^{1}}
7025:{\displaystyle \Gamma }
5329:representation of SU(2)
4437:{\displaystyle \cdots }
3169:{\displaystyle \Gamma }
2873:{\displaystyle \Gamma }
2587:is a Lie subalgebra of
1429:is a Lie subalgebra of
10384:Manifold with boundary
10099:Differential structure
9842:
9818:
9531:
9321:
9292:
9264:
9212:
9179:
9105:
9064:
9031:
8971:
8951:Lie algebra cohomology
8940:
8850:
8763:
8714:
8591:
8498:
8452:
8293:
8200:
8145:
8016:
7939:
7887:
7744:
7707:
7684:, the universal cover
7662:
7631:
7599:
7568:
7537:
7491:
7454:
7395:For the complex case,
7386:
7362:
7333:
7309:
7280:
7253:
7163:
7130:
7084:
7046:
7026:
6998:
6957:(cf. the previous §),
6925:
6847:
6800:
6717:
6599:
6456:
6424:
6316:
6288:
6181:
6085:
6015:
5983:
5900:
5872:
5836:
5800:
5764:
5717:
5693:
5634:
5614:
5594:adjoint representation
5586:
5536:
5465:
5441:
5440:{\displaystyle dc_{g}}
5411:
5341:adjoint representation
5311:. The assumption that
5301:
5273:
5249:
5226:
5201:
5171:
5094:
5034:
4932:
4790:
4727:
4704:
4438:
4415:
4303:
4253:
4208:
4181:
4146:
4116:
4052:
4020:
3988:
3956:
3920:
3892:
3851:one-parameter subgroup
3835:
3807:
3744:
3690:
3650:Example: Each element
3641:
3576:
3539:
3510:
3481:
3432:
3338:
3292:
3262:
3227:
3170:
3150:
3082:
3058:
3021:
2874:
2844:
2815:
2764:
2744:
2743:{\displaystyle Lie(G)}
2709:
2645:
2613:
2581:
2547:
2518:
2482:
2399:
2371:
2309:
2265:
2217:
2158:
2083:
2036:
2013:
1971:
1884:
1850:
1730:
1657:
1607:
1515:
1455:
1423:
1379:
1314:
1282:
1240:
1167:
1095:
1031:), where "exp" is the
1017:
936:
862:Lie group homomorphism
851:
801:
666:#Related constructions
654:
619:
618:{\displaystyle T_{e}G}
573:
533:
493:
454:
423:
345:
300:
258:
91:
62:
10919:Differential geometry
10008:Popov, V.L. (2001) ,
9875:Bourbaki, N. (1981),
9860:, Ch. III, § 3. no. 7
9843:
9819:
9532:
9322:
9293:
9265:
9213:
9185:, the Lie algebra of
9180:
9115:. The Lie algebra of
9106:
9065:
9032:
8997:Related constructions
8972:
8941:
8851:
8764:
8715:
8592:
8499:
8453:
8294:
8201:
8146:
8017:
7940:
7888:
7745:
7708:
7663:
7632:
7600:
7574:is negative definite.
7569:
7543:as a closed subgroup.
7538:
7492:
7455:
7387:
7363:
7334:
7315:whose Lie algebra is
7310:
7281:
7254:
7164:
7131:
7085:
7047:
7045:{\displaystyle \exp }
7027:
6999:
6926:
6848:
6801:
6718:
6600:
6457:
6425:
6317:
6289:
6182:
6086:
6016:
5984:
5901:
5873:
5837:
5806:whose Lie algebra is
5801:
5765:
5723:is determined by the
5718:
5694:
5635:
5615:
5587:
5537:
5466:
5442:
5412:
5302:
5274:
5250:
5227:
5225:{\displaystyle \pi '}
5202:
5200:{\displaystyle d\pi }
5172:
5095:
5035:
4933:
4791:
4728:
4726:{\displaystyle \phi }
4705:
4439:
4416:
4304:
4254:
4209:
4207:{\displaystyle e^{Y}}
4182:
4180:{\displaystyle e^{X}}
4147:
4145:{\displaystyle e^{X}}
4117:
4053:
4021:
3989:
3957:
3955:{\displaystyle \phi }
3921:
3893:
3836:
3808:
3745:
3691:
3642:
3577:
3540:
3511:
3482:
3433:
3339:
3293:
3263:
3228:
3171:
3151:
3083:
3059:
3022:
2875:
2845:
2816:
2765:
2745:
2710:
2646:
2614:
2582:
2548:
2519:
2483:
2400:
2357:
2310:
2266:
2218:
2159:
2084:
2037:
2014:
1972:
1885:
1851:
1731:
1658:
1608:
1537:and if, in addition,
1516:
1456:
1424:
1380:
1315:
1283:
1241:
1168:
1096:
1018:
937:
852:
802:
655:
620:
574:
534:
494:
455:
453:{\displaystyle L_{g}}
424:
346:
301:
259:
99:real coordinate space
92:
63:
10531:Covariant derivative
10082:Topological manifold
9828:
9763:
9431:
9342:Milnor–Moore theorem
9320:{\displaystyle A(G)}
9302:
9278:
9226:
9220:Milnor–Moore theorem
9211:{\displaystyle A(G)}
9193:
9123:
9104:{\displaystyle A(G)}
9086:
9063:{\displaystyle A(G)}
9045:
9009:
8957:
8880:
8773:
8726:
8619:
8508:
8464:
8321:
8210:
8157:
8044:
7949:
7899:
7782:
7721:
7688:
7648:
7609:
7585:
7554:
7504:
7471:
7435:
7372:
7343:
7319:
7290:
7266:
7187:
7144:
7104:
7056:
7036:
7016:
6972:
6857:
6825:
6734:
6610:
6473:
6442:
6330:
6300:
6194:
6131:
6048:
6014:{\displaystyle Z(G)}
5996:
5917:
5886:
5846:
5810:
5774:
5738:
5703:
5644:
5624:
5600:
5546:
5475:
5451:
5421:
5363:
5287:
5259:
5239:
5211:
5188:
5184:. (The differential
5114:
5051:
5003:
4803:
4745:
4717:
4459:
4428:
4316:
4263:
4226:
4191:
4164:
4129:
4065:
4030:
3998:
3966:
3946:
3906:
3861:
3817:
3757:
3703:
3658:
3586:
3557:
3520:
3491:
3450:
3348:
3314:
3276:
3237:
3187:
3160:
3096:
3068:
3035:
2889:
2864:
2828:
2774:
2754:
2719:
2699:
2631:
2591:
2567:
2528:
2496:
2436:
2323:
2287:
2281:general linear group
2231:
2184:
2097:
2049:
2023:
1991:
1894:
1883:{\displaystyle H,H'}
1863:
1743:
1667:
1621:
1565:
1529:is surjective, then
1485:
1433:
1401:
1324:
1292:
1260:
1189:
1112:
1040:
956:
871:
829:
699:
632:
599:
551:
511:
471:
437:
355:
310:
271:
193:
141:connected components
72:
43:
10565:Exterior derivative
10167:Atiyah–Singer index
10116:Riemannian manifold
9416:More generally, if
9337:Compact Lie algebra
7399:are important; see
6696: for all
6578: for all
6025:. If the center of
5952:
5417:; the differential
5107:. The differential
3654:in the Lie algebra
3291:{\displaystyle Lie}
2843:{\displaystyle Lie}
2563:is a Lie group and
2419:Lie's third theorem
1465:is injective, then
775: for all
692:may be computed as
131:. In this article,
10871:Secondary calculus
10825:Singularity theory
10780:Parallel transport
10548:De Rham cohomology
10187:Generalized Stokes
9943:, Academic Press,
9937:Helgason, Sigurdur
9838:
9814:
9527:
9352:Malcev Lie algebra
9317:
9288:
9260:
9208:
9187:primitive elements
9175:
9101:
9070:be the algebra of
9060:
9027:
8979:de Rham cohomology
8967:
8936:
8846:
8759:
8710:
8587:
8494:
8448:
8442:
8289:
8196:
8141:
8012:
7935:
7883:
7756:Compact Lie group
7740:
7703:
7682:with finite center
7658:
7627:
7595:
7564:
7533:
7487:
7467:The adjoint group
7450:
7407:Compact Lie groups
7382:
7358:
7329:
7305:
7276:
7249:
7173:; in other words,
7159:
7126:
7080:
7042:
7022:
7012:and is denoted by
6994:
6935:Abelian Lie groups
6921:
6843:
6796:
6713:
6595:
6452:
6420:
6312:
6284:
6177:
6081:
6011:
5979:
5896:
5868:
5832:
5796:
5760:
5713:
5689:
5630:
5620:and is denoted by
5610:
5582:
5532:
5461:
5437:
5407:
5297:
5283:representation of
5269:
5245:
5222:
5197:
5167:
5090:
5030:
4928:
4786:
4723:
4700:
4434:
4411:
4299:
4249:
4204:
4177:
4142:
4112:
4048:
4016:
3984:
3952:
3916:
3888:
3831:
3803:
3740:
3686:
3637:
3572:
3535:
3506:
3477:
3428:
3334:
3288:
3258:
3223:
3166:
3146:
3078:
3054:
3017:
2870:
2840:
2811:
2760:
2740:
2705:
2641:
2609:
2577:
2543:
2514:
2478:
2409:The correspondence
2395:
2305:
2261:
2213:
2202:
2154:
2079:
2035:{\displaystyle df}
2032:
2009:
1967:
1880:
1846:
1726:
1653:
1617:of Lie groups and
1603:
1511:
1451:
1419:
1389:In particular, if
1375:
1310:
1278:
1236:
1163:
1091:
1013:
932:
847:
797:
650:
615:
569:
529:
489:
450:
419:
341:
296:
254:
87:
58:
10906:
10905:
10788:
10787:
10553:Differential form
10207:Whitney embedding
10141:Differential form
9979:978-1-4419-9981-8
8924:
8867:
8866:
7838:
7700:
7447:
7413:Compact Lie group
7401:complex Lie group
7355:
7302:
7238:
7227:
7179:central extension
7156:
7099:fundamental group
7095:rigidity argument
6941:Abelian Lie group
6705:
6697:
6677:
6587:
6579:
6550:
6021:is the center of
5953:
5248:{\displaystyle G}
4633:
4587:
4373:
4345:
3608:
3569:
3532:
3249:
2763:{\displaystyle f}
2708:{\displaystyle G}
2627:with Lie algebra
2187:
1551:Ehresmann's lemma
1541:is compact, then
776:
672:Matrix Lie groups
122:complex Lie group
120:-adic cases, see
10941:
10898:Stratified space
10856:Fréchet manifold
10570:Interior product
10463:
10462:
10160:
10056:
10049:
10042:
10033:
10032:
10022:
9991:
9953:
9932:
9906:
9880:
9861:
9855:
9849:
9847:
9845:
9844:
9839:
9837:
9836:
9823:
9821:
9820:
9815:
9810:
9809:
9791:
9790:
9781:
9780:
9757:
9751:
9745:
9739:
9733:
9727:
9721:
9715:
9709:
9703:
9698:
9692:
9686:
9680:
9674:
9668:
9662:
9656:
9650:
9644:
9643:Proposition 4.35
9638:
9632:
9626:
9620:
9614:
9608:
9602:
9596:
9590:
9584:
9578:
9572:
9566:
9560:
9554:
9548:
9542:
9536:
9534:
9533:
9528:
9517:
9497:
9496:
9466:
9455:
9454:
9417:
9412:
9403:
9397:
9391:
9385:
9379:
9373:
9326:
9324:
9323:
9318:
9297:
9295:
9294:
9289:
9287:
9286:
9269:
9267:
9266:
9261:
9241:
9240:
9217:
9215:
9214:
9209:
9184:
9182:
9181:
9176:
9132:
9131:
9110:
9108:
9107:
9102:
9069:
9067:
9066:
9061:
9036:
9034:
9033:
9028:
8976:
8974:
8973:
8968:
8966:
8965:
8945:
8943:
8942:
8937:
8926:
8925:
8922:
8910:
8902:
8901:
8892:
8891:
8855:
8853:
8852:
8847:
8845:
8841:
8828:
8827:
8826:
8810:
8802:
8801:
8768:
8766:
8765:
8760:
8755:
8738:
8737:
8719:
8717:
8716:
8711:
8709:
8705:
8674:
8673:
8672:
8656:
8648:
8647:
8596:
8594:
8593:
8588:
8586:
8582:
8563:
8562:
8561:
8545:
8537:
8536:
8503:
8501:
8500:
8495:
8490:
8476:
8475:
8457:
8455:
8454:
8449:
8447:
8446:
8434:
8433:
8417:
8416:
8384:
8380:
8367:
8366:
8365:
8298:
8296:
8295:
8290:
8288:
8284:
8271:
8270:
8269:
8253:
8245:
8244:
8205:
8203:
8202:
8197:
8192:
8169:
8168:
8150:
8148:
8147:
8142:
8140:
8136:
8105:
8104:
8103:
8087:
8079:
8078:
8021:
8019:
8018:
8013:
7987:
7979:
7978:
7944:
7942:
7941:
7936:
7931:
7911:
7910:
7892:
7890:
7889:
7884:
7882:
7878:
7847:
7846:
7845:
7839:
7831:
7822:
7814:
7813:
7760:Complexification
7753:
7752:
7749:
7747:
7746:
7741:
7739:
7738:
7712:
7710:
7709:
7704:
7702:
7701:
7693:
7667:
7665:
7664:
7659:
7657:
7656:
7636:
7634:
7633:
7628:
7604:
7602:
7601:
7596:
7594:
7593:
7573:
7571:
7570:
7565:
7563:
7562:
7542:
7540:
7539:
7534:
7529:
7496:
7494:
7493:
7488:
7486:
7485:
7459:
7457:
7456:
7451:
7449:
7448:
7440:
7403:for this topic.
7391:
7389:
7388:
7383:
7381:
7380:
7367:
7365:
7364:
7359:
7357:
7356:
7348:
7338:
7336:
7335:
7330:
7328:
7327:
7314:
7312:
7311:
7306:
7304:
7303:
7295:
7285:
7283:
7282:
7277:
7275:
7274:
7258:
7256:
7255:
7250:
7239:
7231:
7229:
7228:
7220:
7205:
7204:
7168:
7166:
7165:
7160:
7158:
7157:
7149:
7135:
7133:
7132:
7127:
7116:
7115:
7089:
7087:
7086:
7081:
7070:
7065:
7064:
7051:
7049:
7048:
7043:
7031:
7029:
7028:
7023:
7003:
7001:
7000:
6995:
6987:
6986:
6930:
6928:
6927:
6922:
6905:
6876:
6852:
6850:
6849:
6844:
6805:
6803:
6802:
6797:
6786:
6785:
6784:
6778:
6777:
6755:
6754:
6722:
6720:
6719:
6714:
6706:
6703:
6698:
6695:
6678:
6675:
6622:
6621:
6604:
6602:
6601:
6596:
6588:
6585:
6580:
6577:
6551:
6548:
6519:
6518:
6491:
6490:
6489:
6483:
6482:
6461:
6459:
6458:
6453:
6451:
6450:
6429:
6427:
6426:
6421:
6413:
6412:
6397:
6396:
6342:
6341:
6321:
6319:
6318:
6313:
6293:
6291:
6290:
6285:
6274:
6273:
6258:
6257:
6215:
6214:
6186:
6184:
6183:
6178:
6090:
6088:
6087:
6082:
6020:
6018:
6017:
6012:
5988:
5986:
5985:
5980:
5969:
5968:
5944:
5905:
5903:
5902:
5897:
5895:
5894:
5877:
5875:
5874:
5869:
5864:
5863:
5841:
5839:
5838:
5833:
5828:
5827:
5805:
5803:
5802:
5797:
5792:
5791:
5769:
5767:
5766:
5761:
5756:
5755:
5722:
5720:
5719:
5714:
5712:
5711:
5698:
5696:
5695:
5690:
5639:
5637:
5636:
5631:
5619:
5617:
5616:
5611:
5609:
5608:
5591:
5589:
5588:
5583:
5578:
5577:
5568:
5567:
5555:
5554:
5541:
5539:
5538:
5533:
5531:
5530:
5505:
5504:
5470:
5468:
5467:
5462:
5460:
5459:
5446:
5444:
5443:
5438:
5436:
5435:
5416:
5414:
5413:
5408:
5406:
5405:
5375:
5374:
5359:by conjugation:
5306:
5304:
5303:
5298:
5296:
5295:
5278:
5276:
5275:
5270:
5268:
5267:
5254:
5252:
5251:
5246:
5231:
5229:
5228:
5223:
5221:
5206:
5204:
5203:
5198:
5176:
5174:
5173:
5168:
5160:
5152:
5151:
5146:
5145:
5132:
5131:
5099:
5097:
5096:
5091:
5086:
5078:
5077:
5039:
5037:
5036:
5031:
5026:
5018:
5017:
4937:
4935:
4934:
4929:
4924:
4920:
4919:
4903:
4899:
4898:
4879:
4878:
4860:
4859:
4838:
4834:
4833:
4832:
4823:
4822:
4795:
4793:
4792:
4787:
4785:
4784:
4766:
4765:
4732:
4730:
4729:
4724:
4709:
4707:
4706:
4701:
4696:
4695:
4634:
4626:
4588:
4580:
4540:
4539:
4518:
4514:
4513:
4494:
4490:
4489:
4488:
4479:
4478:
4443:
4441:
4440:
4435:
4420:
4418:
4417:
4412:
4374:
4366:
4346:
4338:
4308:
4306:
4305:
4300:
4298:
4297:
4285:
4284:
4275:
4274:
4258:
4256:
4255:
4250:
4248:
4247:
4238:
4237:
4213:
4211:
4210:
4205:
4203:
4202:
4186:
4184:
4183:
4178:
4176:
4175:
4151:
4149:
4148:
4143:
4141:
4140:
4121:
4119:
4118:
4113:
4108:
4107:
4083:
4082:
4057:
4055:
4054:
4049:
4026:, we may define
4025:
4023:
4022:
4017:
3993:
3991:
3990:
3985:
3961:
3959:
3958:
3953:
3925:
3923:
3922:
3917:
3915:
3914:
3897:
3895:
3894:
3889:
3840:
3838:
3837:
3832:
3824:
3812:
3810:
3809:
3804:
3802:
3794:
3789:
3788:
3773:
3749:
3747:
3746:
3741:
3720:
3719:
3710:
3695:
3693:
3692:
3687:
3667:
3666:
3646:
3644:
3643:
3638:
3636:
3635:
3623:
3622:
3610:
3609:
3601:
3598:
3597:
3581:
3579:
3578:
3573:
3571:
3570:
3562:
3544:
3542:
3541:
3536:
3534:
3533:
3525:
3515:
3513:
3512:
3507:
3505:
3504:
3503:
3486:
3484:
3483:
3478:
3473:
3465:
3464:
3437:
3435:
3434:
3429:
3421:
3413:
3412:
3385:
3377:
3376:
3371:
3370:
3357:
3356:
3343:
3341:
3340:
3335:
3333:
3332:
3327:
3326:
3297:
3295:
3294:
3289:
3267:
3265:
3264:
3259:
3251:
3250:
3242:
3232:
3230:
3229:
3224:
3175:
3173:
3172:
3167:
3155:
3153:
3152:
3147:
3139:
3138:
3111:
3110:
3087:
3085:
3084:
3079:
3077:
3076:
3063:
3061:
3060:
3055:
3050:
3049:
3026:
3024:
3023:
3018:
2989:
2988:
2979:
2978:
2964:
2937:
2936:
2921:
2920:
2903:
2879:
2877:
2876:
2871:
2849:
2847:
2846:
2841:
2820:
2818:
2817:
2812:
2810:
2809:
2769:
2767:
2766:
2761:
2749:
2747:
2746:
2741:
2715:its Lie algebra
2714:
2712:
2711:
2706:
2650:
2648:
2647:
2642:
2640:
2639:
2618:
2616:
2615:
2610:
2586:
2584:
2583:
2578:
2576:
2575:
2552:
2550:
2549:
2544:
2523:
2521:
2520:
2515:
2487:
2485:
2484:
2479:
2404:
2402:
2401:
2396:
2394:
2387:
2382:
2381:
2370:
2365:
2353:
2352:
2314:
2312:
2311:
2306:
2270:
2268:
2267:
2262:
2222:
2220:
2219:
2214:
2212:
2211:
2201:
2163:
2161:
2160:
2155:
2088:
2086:
2085:
2080:
2041:
2039:
2038:
2033:
2018:
2016:
2015:
2010:
1976:
1974:
1973:
1968:
1960:
1919:
1889:
1887:
1886:
1881:
1879:
1855:
1853:
1852:
1847:
1839:
1838:
1808:
1807:
1783:
1782:
1764:
1763:
1735:
1733:
1732:
1727:
1722:
1721:
1682:
1681:
1662:
1660:
1659:
1654:
1652:
1651:
1633:
1632:
1612:
1610:
1609:
1604:
1602:
1601:
1583:
1582:
1557:Other properties
1547:principal bundle
1520:
1518:
1517:
1512:
1495:
1460:
1458:
1457:
1452:
1428:
1426:
1425:
1420:
1384:
1382:
1381:
1376:
1319:
1317:
1316:
1311:
1287:
1285:
1284:
1279:
1245:
1243:
1242:
1237:
1199:
1172:
1170:
1169:
1164:
1108:is closed, then
1104:If the image of
1100:
1098:
1097:
1092:
1022:
1020:
1019:
1014:
941:
939:
938:
933:
895:
894:
856:
854:
853:
848:
806:
804:
803:
798:
793:
789:
788:
777:
774:
766:
765:
747:
659:
657:
656:
651:
624:
622:
621:
616:
611:
610:
578:
576:
575:
570:
538:
536:
535:
532:{\displaystyle }
530:
498:
496:
495:
490:
459:
457:
456:
451:
449:
448:
428:
426:
425:
420:
415:
414:
396:
395:
383:
382:
373:
372:
350:
348:
347:
342:
322:
321:
305:
303:
302:
297:
283:
282:
263:
261:
260:
255:
253:
252:
234:
233:
221:
220:
211:
210:
137:second countable
107:simply connected
96:
94:
93:
88:
86:
85:
80:
67:
65:
64:
59:
57:
56:
51:
10949:
10948:
10944:
10943:
10942:
10940:
10939:
10938:
10909:
10908:
10907:
10902:
10841:Banach manifold
10834:Generalizations
10829:
10784:
10721:
10618:
10580:Ricci curvature
10536:Cotangent space
10514:
10452:
10294:
10288:
10247:Exponential map
10211:
10156:
10150:
10070:
10060:
9999:
9994:
9980:
9970:Springer-Verlag
9951:
9930:
9904:
9870:
9865:
9864:
9856:
9852:
9832:
9831:
9829:
9826:
9825:
9805:
9804:
9786:
9782:
9776:
9775:
9764:
9761:
9760:
9758:
9754:
9746:
9742:
9734:
9730:
9722:
9718:
9710:
9706:
9699:
9695:
9687:
9683:
9675:
9671:
9663:
9659:
9651:
9647:
9639:
9635:
9627:
9623:
9615:
9611:
9603:
9599:
9591:
9587:
9579:
9575:
9567:
9563:
9555:
9551:
9543:
9539:
9510:
9489:
9485:
9459:
9447:
9443:
9432:
9429:
9428:
9415:
9406:
9398:
9394:
9386:
9382:
9374:
9370:
9365:
9333:
9303:
9300:
9299:
9282:
9281:
9279:
9276:
9275:
9236:
9235:
9227:
9224:
9223:
9194:
9191:
9190:
9127:
9126:
9124:
9121:
9120:
9087:
9084:
9083:
9046:
9043:
9042:
9010:
9007:
9006:
8999:
8961:
8960:
8958:
8955:
8954:
8921:
8917:
8906:
8897:
8896:
8887:
8883:
8881:
8878:
8877:
8863:
8822:
8821:
8817:
8806:
8794:
8790:
8783:
8779:
8774:
8771:
8770:
8751:
8730:
8729:
8727:
8724:
8723:
8668:
8667:
8663:
8652:
8640:
8636:
8629:
8625:
8620:
8617:
8616:
8604:
8557:
8556:
8552:
8541:
8529:
8525:
8518:
8514:
8509:
8506:
8505:
8486:
8468:
8467:
8465:
8462:
8461:
8441:
8440:
8435:
8429:
8425:
8419:
8418:
8412:
8408:
8406:
8396:
8395:
8361:
8360:
8356:
8331:
8327:
8322:
8319:
8318:
8306:
8265:
8264:
8260:
8249:
8231:
8227:
8220:
8216:
8211:
8208:
8207:
8188:
8161:
8160:
8158:
8155:
8154:
8099:
8098:
8094:
8083:
8065:
8061:
8054:
8050:
8045:
8042:
8041:
8029:
7983:
7968:
7964:
7950:
7947:
7946:
7927:
7903:
7902:
7900:
7897:
7896:
7841:
7840:
7830:
7829:
7818:
7803:
7799:
7792:
7788:
7783:
7780:
7779:
7734:
7730:
7722:
7719:
7718:
7692:
7691:
7689:
7686:
7685:
7652:
7651:
7649:
7646:
7645:
7610:
7607:
7606:
7589:
7588:
7586:
7583:
7582:
7558:
7557:
7555:
7552:
7551:
7525:
7505:
7502:
7501:
7481:
7480:
7472:
7469:
7468:
7439:
7438:
7436:
7433:
7432:
7415:
7409:
7376:
7375:
7373:
7370:
7369:
7347:
7346:
7344:
7341:
7340:
7323:
7322:
7320:
7317:
7316:
7294:
7293:
7291:
7288:
7287:
7270:
7269:
7267:
7264:
7263:
7230:
7219:
7218:
7200:
7196:
7188:
7185:
7184:
7148:
7147:
7145:
7142:
7141:
7111:
7107:
7105:
7102:
7101:
7066:
7060:
7059:
7057:
7054:
7053:
7037:
7034:
7033:
7017:
7014:
7013:
7006:integer lattice
6982:
6981:
6973:
6970:
6969:
6943:
6937:
6901:
6872:
6858:
6855:
6854:
6826:
6823:
6822:
6780:
6779:
6773:
6772:
6771:
6750:
6746:
6735:
6732:
6731:
6702:
6694:
6674:
6617:
6613:
6611:
6608:
6607:
6584:
6576:
6547:
6514:
6513:
6485:
6484:
6478:
6477:
6476:
6474:
6471:
6470:
6446:
6445:
6443:
6440:
6439:
6408:
6404:
6392:
6388:
6337:
6333:
6331:
6328:
6327:
6301:
6298:
6297:
6269:
6265:
6253:
6249:
6210:
6206:
6195:
6192:
6191:
6132:
6129:
6128:
6118:
6049:
6046:
6045:
6044:if and only if
5997:
5994:
5993:
5964:
5963:
5918:
5915:
5914:
5890:
5889:
5887:
5884:
5883:
5859:
5858:
5847:
5844:
5843:
5823:
5822:
5811:
5808:
5807:
5787:
5786:
5775:
5772:
5771:
5751:
5750:
5739:
5736:
5735:
5707:
5706:
5704:
5701:
5700:
5645:
5642:
5641:
5640:. One can show
5625:
5622:
5621:
5604:
5603:
5601:
5598:
5597:
5573:
5572:
5560:
5559:
5550:
5549:
5547:
5544:
5543:
5526:
5522:
5500:
5499:
5476:
5473:
5472:
5455:
5454:
5452:
5449:
5448:
5431:
5427:
5422:
5419:
5418:
5398:
5394:
5370:
5366:
5364:
5361:
5360:
5351:in a Lie group
5347:; each element
5343:of a Lie group
5337:
5291:
5290:
5288:
5285:
5284:
5263:
5262:
5260:
5257:
5256:
5240:
5237:
5236:
5214:
5212:
5209:
5208:
5189:
5186:
5185:
5156:
5147:
5138:
5137:
5136:
5127:
5126:
5115:
5112:
5111:
5082:
5073:
5069:
5052:
5049:
5048:
5022:
5013:
5009:
5004:
5001:
5000:
4990:
4915:
4911:
4907:
4894:
4890:
4886:
4865:
4861:
4846:
4842:
4828:
4824:
4818:
4814:
4813:
4809:
4804:
4801:
4800:
4771:
4767:
4752:
4748:
4746:
4743:
4742:
4718:
4715:
4714:
4625:
4579:
4548:
4544:
4526:
4522:
4509:
4505:
4501:
4484:
4480:
4474:
4470:
4469:
4465:
4460:
4457:
4456:
4429:
4426:
4425:
4365:
4337:
4317:
4314:
4313:
4293:
4289:
4280:
4276:
4270:
4266:
4264:
4261:
4260:
4243:
4239:
4233:
4229:
4227:
4224:
4223:
4198:
4194:
4192:
4189:
4188:
4171:
4167:
4165:
4162:
4161:
4136:
4132:
4130:
4127:
4126:
4094:
4090:
4078:
4074:
4066:
4063:
4062:
4031:
4028:
4027:
3999:
3996:
3995:
3967:
3964:
3963:
3947:
3944:
3943:
3936:
3910:
3909:
3907:
3904:
3903:
3862:
3859:
3858:
3820:
3818:
3815:
3814:
3798:
3790:
3784:
3780:
3769:
3758:
3755:
3754:
3715:
3714:
3706:
3704:
3701:
3700:
3662:
3661:
3659:
3656:
3655:
3631:
3630:
3618:
3614:
3600:
3599:
3593:
3589:
3587:
3584:
3583:
3561:
3560:
3558:
3555:
3554:
3524:
3523:
3521:
3518:
3517:
3499:
3498:
3494:
3492:
3489:
3488:
3469:
3460:
3456:
3451:
3448:
3447:
3417:
3408:
3404:
3381:
3372:
3363:
3362:
3361:
3352:
3351:
3349:
3346:
3345:
3328:
3319:
3318:
3317:
3315:
3312:
3311:
3304:
3277:
3274:
3273:
3241:
3240:
3238:
3235:
3234:
3188:
3185:
3184:
3161:
3158:
3157:
3134:
3133:
3106:
3105:
3097:
3094:
3093:
3072:
3071:
3069:
3066:
3065:
3045:
3044:
3036:
3033:
3032:
2984:
2983:
2965:
2954:
2953:
2932:
2931:
2904:
2893:
2892:
2890:
2887:
2886:
2865:
2862:
2861:
2859:adjoint functor
2829:
2826:
2825:
2805:
2801:
2775:
2772:
2771:
2755:
2752:
2751:
2720:
2717:
2716:
2700:
2697:
2696:
2689:category theory
2635:
2634:
2632:
2629:
2628:
2592:
2589:
2588:
2571:
2570:
2568:
2565:
2564:
2529:
2526:
2525:
2497:
2494:
2493:
2437:
2434:
2433:
2411:
2383:
2377:
2373:
2372:
2366:
2361:
2348:
2344:
2324:
2321:
2320:
2288:
2285:
2284:
2232:
2229:
2228:
2223:coincides with
2207:
2203:
2191:
2185:
2182:
2181:
2098:
2095:
2094:
2050:
2047:
2046:
2044:exponential map
2024:
2021:
2020:
1992:
1989:
1988:
1953:
1912:
1895:
1892:
1891:
1872:
1864:
1861:
1860:
1834:
1830:
1803:
1799:
1778:
1774:
1759:
1755:
1744:
1741:
1740:
1717:
1713:
1677:
1673:
1668:
1665:
1664:
1647:
1643:
1628:
1624:
1622:
1619:
1618:
1597:
1593:
1578:
1574:
1566:
1563:
1562:
1559:
1491:
1486:
1483:
1482:
1481:. For example,
1434:
1431:
1430:
1402:
1399:
1398:
1325:
1322:
1321:
1293:
1290:
1289:
1261:
1258:
1257:
1195:
1190:
1187:
1186:
1113:
1110:
1109:
1041:
1038:
1037:
1033:exponential map
957:
954:
953:
890:
886:
872:
869:
868:
830:
827:
826:
820:
784:
773:
758:
754:
743:
724:
720:
700:
697:
696:
674:
633:
630:
629:
606:
602:
600:
597:
596:
552:
549:
548:
512:
509:
508:
472:
469:
468:
444:
440:
438:
435:
434:
407:
403:
391:
387:
378:
374:
368:
364:
356:
353:
352:
317:
313:
311:
308:
307:
278:
274:
272:
269:
268:
245:
241:
229:
225:
216:
212:
206:
202:
194:
191:
190:
154:
149:
129:-adic Lie group
81:
76:
75:
73:
70:
69:
52:
47:
46:
44:
41:
40:
17:
12:
11:
5:
10947:
10937:
10936:
10931:
10926:
10921:
10904:
10903:
10901:
10900:
10895:
10890:
10885:
10880:
10879:
10878:
10868:
10863:
10858:
10853:
10848:
10843:
10837:
10835:
10831:
10830:
10828:
10827:
10822:
10817:
10812:
10807:
10802:
10796:
10794:
10790:
10789:
10786:
10785:
10783:
10782:
10777:
10772:
10767:
10762:
10757:
10752:
10747:
10742:
10737:
10731:
10729:
10723:
10722:
10720:
10719:
10714:
10709:
10704:
10699:
10694:
10689:
10679:
10674:
10669:
10659:
10654:
10649:
10644:
10639:
10634:
10628:
10626:
10620:
10619:
10617:
10616:
10611:
10606:
10605:
10604:
10594:
10589:
10588:
10587:
10577:
10572:
10567:
10562:
10561:
10560:
10550:
10545:
10544:
10543:
10533:
10528:
10522:
10520:
10516:
10515:
10513:
10512:
10507:
10502:
10497:
10496:
10495:
10485:
10480:
10475:
10469:
10467:
10460:
10454:
10453:
10451:
10450:
10445:
10435:
10430:
10416:
10411:
10406:
10401:
10396:
10394:Parallelizable
10391:
10386:
10381:
10380:
10379:
10369:
10364:
10359:
10354:
10349:
10344:
10339:
10334:
10329:
10324:
10314:
10304:
10298:
10296:
10290:
10289:
10287:
10286:
10281:
10276:
10274:Lie derivative
10271:
10269:Integral curve
10266:
10261:
10256:
10255:
10254:
10244:
10239:
10238:
10237:
10230:Diffeomorphism
10227:
10221:
10219:
10213:
10212:
10210:
10209:
10204:
10199:
10194:
10189:
10184:
10179:
10174:
10169:
10163:
10161:
10152:
10151:
10149:
10148:
10143:
10138:
10133:
10128:
10123:
10118:
10113:
10108:
10107:
10106:
10101:
10091:
10090:
10089:
10078:
10076:
10075:Basic concepts
10072:
10071:
10059:
10058:
10051:
10044:
10036:
10030:
10029:
10023:
10005:
9998:
9997:External links
9995:
9993:
9992:
9978:
9954:
9949:
9933:
9929:978-3319134666
9928:
9907:
9902:
9881:
9871:
9869:
9866:
9863:
9862:
9850:
9835:
9813:
9808:
9803:
9800:
9797:
9794:
9789:
9785:
9779:
9774:
9771:
9768:
9752:
9740:
9728:
9716:
9704:
9693:
9681:
9669:
9657:
9645:
9633:
9621:
9609:
9597:
9585:
9583:Corollary 3.49
9573:
9561:
9549:
9537:
9526:
9523:
9520:
9516:
9513:
9509:
9506:
9503:
9500:
9495:
9492:
9488:
9484:
9481:
9478:
9475:
9472:
9469:
9465:
9462:
9458:
9453:
9450:
9446:
9442:
9439:
9436:
9404:
9392:
9380:
9378:, p. 530.
9367:
9366:
9364:
9361:
9360:
9359:
9354:
9349:
9344:
9339:
9332:
9329:
9316:
9313:
9310:
9307:
9285:
9259:
9256:
9253:
9250:
9247:
9244:
9239:
9234:
9231:
9207:
9204:
9201:
9198:
9174:
9171:
9168:
9165:
9162:
9159:
9156:
9153:
9150:
9147:
9144:
9141:
9138:
9135:
9130:
9100:
9097:
9094:
9091:
9059:
9056:
9053:
9050:
9026:
9023:
9020:
9017:
9014:
8998:
8995:
8991:left invariant
8964:
8947:
8946:
8935:
8932:
8929:
8920:
8916:
8913:
8909:
8905:
8900:
8895:
8890:
8886:
8865:
8864:
8861:
8856:
8844:
8840:
8837:
8834:
8831:
8825:
8820:
8816:
8813:
8809:
8805:
8800:
8797:
8793:
8789:
8786:
8782:
8778:
8758:
8754:
8750:
8747:
8744:
8741:
8736:
8733:
8720:
8708:
8704:
8701:
8698:
8695:
8692:
8689:
8686:
8683:
8680:
8677:
8671:
8666:
8662:
8659:
8655:
8651:
8646:
8643:
8639:
8635:
8632:
8628:
8624:
8606:
8605:
8602:
8597:
8585:
8581:
8578:
8575:
8572:
8569:
8566:
8560:
8555:
8551:
8548:
8544:
8540:
8535:
8532:
8528:
8524:
8521:
8517:
8513:
8493:
8489:
8485:
8482:
8479:
8474:
8471:
8458:
8445:
8439:
8436:
8432:
8428:
8424:
8421:
8420:
8415:
8411:
8407:
8405:
8402:
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8399:
8394:
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8387:
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8373:
8370:
8364:
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8346:
8343:
8340:
8337:
8334:
8330:
8326:
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8299:
8287:
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8280:
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8263:
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8230:
8226:
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8114:
8111:
8108:
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8077:
8074:
8071:
8068:
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8049:
8031:
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8027:
8022:
8011:
8008:
8005:
8002:
7999:
7996:
7993:
7990:
7986:
7982:
7977:
7974:
7971:
7967:
7963:
7960:
7957:
7954:
7934:
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7926:
7923:
7920:
7917:
7914:
7909:
7906:
7893:
7881:
7877:
7874:
7871:
7868:
7865:
7862:
7859:
7856:
7853:
7850:
7844:
7837:
7834:
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7821:
7817:
7812:
7809:
7806:
7802:
7798:
7795:
7791:
7787:
7769:
7768:
7763:
7757:
7737:
7733:
7729:
7726:
7699:
7696:
7670:
7669:
7655:
7642:
7639:diagonalizable
7626:
7623:
7620:
7617:
7614:
7592:
7575:
7561:
7544:
7532:
7528:
7524:
7521:
7518:
7515:
7512:
7509:
7498:
7484:
7479:
7476:
7465:
7446:
7443:
7429:
7411:Main article:
7408:
7405:
7379:
7354:
7351:
7326:
7301:
7298:
7273:
7260:
7259:
7248:
7245:
7242:
7237:
7234:
7226:
7223:
7217:
7214:
7211:
7208:
7203:
7199:
7195:
7192:
7177:fits into the
7155:
7152:
7125:
7122:
7119:
7114:
7110:
7079:
7076:
7073:
7069:
7063:
7041:
7021:
6993:
6990:
6985:
6980:
6977:
6939:Main article:
6936:
6933:
6920:
6917:
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6911:
6908:
6904:
6900:
6897:
6894:
6891:
6888:
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6761:
6758:
6753:
6749:
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6724:
6723:
6712:
6709:
6704: in
6701:
6693:
6690:
6687:
6684:
6681:
6676: or
6673:
6670:
6667:
6664:
6661:
6658:
6655:
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6643:
6640:
6637:
6634:
6631:
6628:
6625:
6620:
6616:
6605:
6594:
6591:
6586: in
6583:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6554:
6549: or
6546:
6543:
6540:
6537:
6534:
6531:
6528:
6525:
6522:
6517:
6512:
6509:
6506:
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6500:
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6494:
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6449:
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6419:
6416:
6411:
6407:
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6400:
6395:
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6387:
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6360:
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6340:
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6280:
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6264:
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6227:
6224:
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6213:
6209:
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6199:
6176:
6173:
6170:
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6114:
6080:
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6010:
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5990:
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5978:
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5951:
5947:
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5925:
5922:
5893:
5867:
5862:
5857:
5854:
5851:
5831:
5826:
5821:
5818:
5815:
5795:
5790:
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5779:
5759:
5754:
5749:
5746:
5743:
5710:
5688:
5685:
5682:
5679:
5676:
5673:
5670:
5667:
5664:
5661:
5658:
5655:
5652:
5649:
5629:
5607:
5592:is called the
5581:
5576:
5571:
5566:
5563:
5558:
5553:
5529:
5525:
5521:
5518:
5515:
5511:
5508:
5503:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5458:
5434:
5430:
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5404:
5401:
5397:
5393:
5390:
5387:
5384:
5381:
5378:
5373:
5369:
5336:
5333:
5294:
5266:
5244:
5220:
5217:
5196:
5193:
5178:
5177:
5166:
5163:
5159:
5155:
5150:
5144:
5141:
5135:
5130:
5125:
5122:
5119:
5101:
5100:
5089:
5085:
5081:
5076:
5072:
5068:
5065:
5062:
5059:
5056:
5029:
5025:
5021:
5016:
5012:
5008:
4989:
4986:
4939:
4938:
4927:
4923:
4918:
4914:
4910:
4906:
4902:
4897:
4893:
4889:
4885:
4882:
4877:
4874:
4871:
4868:
4864:
4858:
4855:
4852:
4849:
4845:
4841:
4837:
4831:
4827:
4821:
4817:
4812:
4808:
4783:
4780:
4777:
4774:
4770:
4764:
4761:
4758:
4755:
4751:
4722:
4711:
4710:
4699:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4637:
4632:
4629:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4586:
4583:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4547:
4543:
4538:
4535:
4532:
4529:
4525:
4521:
4517:
4512:
4508:
4504:
4500:
4497:
4493:
4487:
4483:
4477:
4473:
4468:
4464:
4433:
4422:
4421:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4372:
4369:
4364:
4361:
4358:
4355:
4352:
4349:
4344:
4341:
4336:
4333:
4330:
4327:
4324:
4321:
4296:
4292:
4288:
4283:
4279:
4273:
4269:
4246:
4242:
4236:
4232:
4201:
4197:
4174:
4170:
4139:
4135:
4123:
4122:
4111:
4106:
4103:
4100:
4097:
4093:
4089:
4086:
4081:
4077:
4073:
4070:
4047:
4044:
4041:
4038:
4035:
4015:
4012:
4009:
4006:
4003:
3983:
3980:
3977:
3974:
3971:
3951:
3935:
3932:
3913:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3830:
3827:
3823:
3801:
3797:
3793:
3787:
3783:
3779:
3776:
3772:
3768:
3765:
3762:
3751:
3750:
3739:
3736:
3733:
3730:
3727:
3723:
3718:
3713:
3709:
3685:
3682:
3679:
3676:
3673:
3670:
3665:
3634:
3629:
3626:
3621:
3617:
3613:
3607:
3604:
3596:
3592:
3568:
3565:
3531:
3528:
3502:
3497:
3476:
3472:
3468:
3463:
3459:
3455:
3446:) subgroup of
3427:
3424:
3420:
3416:
3411:
3407:
3403:
3400:
3397:
3394:
3391:
3388:
3384:
3380:
3375:
3369:
3366:
3360:
3355:
3331:
3325:
3322:
3303:
3300:
3287:
3284:
3281:
3257:
3254:
3248:
3245:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3178:fully faithful
3165:
3145:
3142:
3137:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3109:
3104:
3101:
3075:
3053:
3048:
3043:
3040:
3030:
3029:
3028:
3027:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2987:
2982:
2977:
2974:
2971:
2968:
2963:
2960:
2957:
2952:
2949:
2946:
2943:
2940:
2935:
2930:
2927:
2924:
2919:
2916:
2913:
2910:
2907:
2902:
2899:
2896:
2869:
2839:
2836:
2833:
2808:
2804:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2759:
2739:
2736:
2733:
2730:
2727:
2724:
2704:
2680:is to use the
2653:
2652:
2638:
2608:
2605:
2602:
2599:
2596:
2574:
2554:
2542:
2539:
2536:
2533:
2513:
2510:
2507:
2504:
2501:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2427:
2410:
2407:
2393:
2390:
2386:
2380:
2376:
2369:
2364:
2360:
2356:
2351:
2347:
2343:
2340:
2337:
2334:
2331:
2328:
2304:
2301:
2298:
2295:
2292:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2210:
2206:
2200:
2197:
2194:
2190:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2031:
2028:
2008:
2005:
2002:
1999:
1996:
1966:
1963:
1959:
1956:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1918:
1915:
1911:
1908:
1905:
1902:
1899:
1878:
1875:
1871:
1868:
1857:
1856:
1845:
1842:
1837:
1833:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1806:
1802:
1798:
1795:
1792:
1789:
1786:
1781:
1777:
1773:
1770:
1767:
1762:
1758:
1754:
1751:
1748:
1725:
1720:
1716:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1680:
1676:
1672:
1650:
1646:
1642:
1639:
1636:
1631:
1627:
1615:direct product
1600:
1596:
1592:
1589:
1586:
1581:
1577:
1573:
1570:
1558:
1555:
1510:
1507:
1504:
1501:
1498:
1494:
1490:
1450:
1447:
1444:
1441:
1438:
1418:
1415:
1412:
1409:
1406:
1387:
1386:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1309:
1306:
1303:
1300:
1297:
1277:
1274:
1271:
1268:
1265:
1250:
1249:
1248:
1247:
1246:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1198:
1194:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1102:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1035:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
943:
942:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
893:
889:
885:
882:
879:
876:
858:
857:
846:
843:
840:
837:
834:
819:
816:
808:
807:
796:
792:
787:
783:
780:
772:
769:
764:
761:
757:
753:
750:
746:
742:
739:
736:
733:
730:
727:
723:
719:
716:
713:
710:
707:
704:
673:
670:
649:
646:
643:
640:
637:
614:
609:
605:
568:
565:
562:
559:
556:
528:
525:
522:
519:
516:
488:
485:
482:
479:
476:
462:tangent spaces
447:
443:
418:
413:
410:
406:
402:
399:
394:
390:
386:
381:
377:
371:
367:
363:
360:
340:
337:
334:
331:
328:
325:
320:
316:
306:is defined by
295:
292:
289:
286:
281:
277:
265:
264:
251:
248:
244:
240:
237:
232:
228:
224:
219:
215:
209:
205:
201:
198:
153:
150:
148:
145:
84:
79:
55:
50:
15:
9:
6:
4:
3:
2:
10946:
10935:
10932:
10930:
10927:
10925:
10922:
10920:
10917:
10916:
10914:
10899:
10896:
10894:
10893:Supermanifold
10891:
10889:
10886:
10884:
10881:
10877:
10874:
10873:
10872:
10869:
10867:
10864:
10862:
10859:
10857:
10854:
10852:
10849:
10847:
10844:
10842:
10839:
10838:
10836:
10832:
10826:
10823:
10821:
10818:
10816:
10813:
10811:
10808:
10806:
10803:
10801:
10798:
10797:
10795:
10791:
10781:
10778:
10776:
10773:
10771:
10768:
10766:
10763:
10761:
10758:
10756:
10753:
10751:
10748:
10746:
10743:
10741:
10738:
10736:
10733:
10732:
10730:
10728:
10724:
10718:
10715:
10713:
10710:
10708:
10705:
10703:
10700:
10698:
10695:
10693:
10690:
10688:
10684:
10680:
10678:
10675:
10673:
10670:
10668:
10664:
10660:
10658:
10655:
10653:
10650:
10648:
10645:
10643:
10640:
10638:
10635:
10633:
10630:
10629:
10627:
10625:
10621:
10615:
10614:Wedge product
10612:
10610:
10607:
10603:
10600:
10599:
10598:
10595:
10593:
10590:
10586:
10583:
10582:
10581:
10578:
10576:
10573:
10571:
10568:
10566:
10563:
10559:
10558:Vector-valued
10556:
10555:
10554:
10551:
10549:
10546:
10542:
10539:
10538:
10537:
10534:
10532:
10529:
10527:
10524:
10523:
10521:
10517:
10511:
10508:
10506:
10503:
10501:
10498:
10494:
10491:
10490:
10489:
10488:Tangent space
10486:
10484:
10481:
10479:
10476:
10474:
10471:
10470:
10468:
10464:
10461:
10459:
10455:
10449:
10446:
10444:
10440:
10436:
10434:
10431:
10429:
10425:
10421:
10417:
10415:
10412:
10410:
10407:
10405:
10402:
10400:
10397:
10395:
10392:
10390:
10387:
10385:
10382:
10378:
10375:
10374:
10373:
10370:
10368:
10365:
10363:
10360:
10358:
10355:
10353:
10350:
10348:
10345:
10343:
10340:
10338:
10335:
10333:
10330:
10328:
10325:
10323:
10319:
10315:
10313:
10309:
10305:
10303:
10300:
10299:
10297:
10291:
10285:
10282:
10280:
10277:
10275:
10272:
10270:
10267:
10265:
10262:
10260:
10257:
10253:
10252:in Lie theory
10250:
10249:
10248:
10245:
10243:
10240:
10236:
10233:
10232:
10231:
10228:
10226:
10223:
10222:
10220:
10218:
10214:
10208:
10205:
10203:
10200:
10198:
10195:
10193:
10190:
10188:
10185:
10183:
10180:
10178:
10175:
10173:
10170:
10168:
10165:
10164:
10162:
10159:
10155:Main results
10153:
10147:
10144:
10142:
10139:
10137:
10136:Tangent space
10134:
10132:
10129:
10127:
10124:
10122:
10119:
10117:
10114:
10112:
10109:
10105:
10102:
10100:
10097:
10096:
10095:
10092:
10088:
10085:
10084:
10083:
10080:
10079:
10077:
10073:
10068:
10064:
10057:
10052:
10050:
10045:
10043:
10038:
10037:
10034:
10027:
10024:
10021:
10017:
10016:
10011:
10006:
10004:
10001:
10000:
9989:
9985:
9981:
9975:
9971:
9967:
9963:
9959:
9955:
9952:
9950:0-12-338460-5
9946:
9942:
9938:
9934:
9931:
9925:
9921:
9917:
9913:
9908:
9905:
9899:
9895:
9891:
9887:
9882:
9878:
9873:
9872:
9859:
9858:Bourbaki 1981
9854:
9798:
9795:
9792:
9787:
9769:
9766:
9756:
9749:
9748:Bourbaki 1981
9744:
9737:
9736:Bourbaki 1981
9732:
9725:
9724:Helgason 1978
9720:
9714:, Section 4.7
9713:
9708:
9702:
9697:
9690:
9685:
9678:
9673:
9667:Corollary 5.7
9666:
9661:
9654:
9649:
9642:
9637:
9630:
9625:
9618:
9613:
9606:
9601:
9594:
9589:
9582:
9577:
9570:
9569:Bourbaki 1981
9565:
9558:
9557:Bourbaki 1981
9553:
9547:
9541:
9524:
9514:
9511:
9504:
9501:
9493:
9490:
9482:
9479:
9473:
9463:
9460:
9451:
9448:
9444:
9437:
9434:
9426:
9422:
9419:
9418:
9411:
9410:
9401:
9396:
9389:
9388:Helgason 1978
9384:
9377:
9372:
9368:
9358:
9355:
9353:
9350:
9348:
9345:
9343:
9340:
9338:
9335:
9334:
9328:
9311:
9305:
9273:
9254:
9248:
9245:
9229:
9221:
9202:
9196:
9188:
9166:
9160:
9154:
9151:
9145:
9139:
9136:
9133:
9118:
9114:
9111:is in fact a
9095:
9089:
9081:
9077:
9073:
9072:distributions
9054:
9048:
9040:
9021:
9015:
9012:
9004:
8994:
8992:
8988:
8984:
8980:
8952:
8930:
8918:
8914:
8903:
8888:
8884:
8876:
8875:
8874:
8872:
8860:
8857:
8842:
8838:
8835:
8832:
8829:
8818:
8814:
8798:
8795:
8791:
8787:
8784:
8780:
8776:
8769:
8748:
8745:
8742:
8721:
8706:
8702:
8699:
8693:
8684:
8681:
8678:
8675:
8664:
8660:
8644:
8641:
8637:
8633:
8630:
8626:
8622:
8615:
8613:
8608:
8607:
8601:
8598:
8583:
8579:
8576:
8573:
8570:
8567:
8564:
8553:
8549:
8533:
8530:
8526:
8522:
8519:
8515:
8511:
8504:
8483:
8480:
8459:
8443:
8437:
8430:
8426:
8422:
8413:
8409:
8403:
8397:
8392:
8389:
8385:
8381:
8377:
8374:
8371:
8368:
8357:
8353:
8347:
8344:
8338:
8335:
8332:
8328:
8324:
8317:
8315:
8310:
8309:
8303:
8300:
8285:
8281:
8278:
8275:
8272:
8261:
8257:
8241:
8238:
8235:
8232:
8228:
8224:
8221:
8217:
8213:
8206:
8185:
8182:
8179:
8176:
8173:
8152:
8137:
8133:
8130:
8124:
8115:
8112:
8109:
8106:
8095:
8091:
8075:
8072:
8069:
8066:
8062:
8058:
8055:
8051:
8047:
8040:
8038:
8033:
8032:
8026:
8023:
8006:
8003:
8000:
7997:
7994:
7991:
7975:
7972:
7969:
7965:
7961:
7958:
7952:
7945:
7924:
7921:
7918:
7915:
7894:
7879:
7875:
7872:
7866:
7857:
7854:
7851:
7848:
7832:
7826:
7810:
7807:
7804:
7800:
7796:
7793:
7789:
7785:
7778:
7776:
7771:
7770:
7767:
7764:
7761:
7758:
7755:
7754:
7751:
7735:
7731:
7727:
7724:
7716:
7697:
7694:
7683:
7679:
7675:
7643:
7640:
7621:
7615:
7612:
7580:
7576:
7549:
7545:
7522:
7519:
7513:
7507:
7499:
7477:
7474:
7466:
7463:
7444:
7441:
7430:
7427:
7424:
7423:
7422:
7420:
7414:
7404:
7402:
7398:
7393:
7352:
7349:
7299:
7296:
7246:
7240:
7235:
7224:
7221:
7209:
7201:
7197:
7190:
7183:
7182:
7181:
7180:
7176:
7172:
7153:
7150:
7139:
7120:
7112:
7108:
7100:
7096:
7091:
7077:
7067:
7039:
7011:
7007:
6991:
6978:
6975:
6967:
6962:
6960:
6956:
6952:
6948:
6942:
6932:
6915:
6909:
6906:
6902:
6895:
6889:
6886:
6883:
6877:
6873:
6869:
6863:
6860:
6837:
6831:
6828:
6820:
6816:
6812:
6807:
6790:
6768:
6759:
6751:
6747:
6740:
6737:
6729:
6707:
6699:
6691:
6688:
6685:
6682:
6679:
6671:
6668:
6665:
6659:
6653:
6650:
6647:
6644:
6641:
6638:
6632:
6626:
6618:
6614:
6606:
6589:
6581:
6573:
6570:
6567:
6561:
6555:
6552:
6544:
6541:
6538:
6532:
6526:
6523:
6520:
6510:
6507:
6501:
6495:
6469:
6468:
6467:
6465:
6437:
6434:For a subset
6414:
6409:
6405:
6398:
6393:
6389:
6385:
6379:
6373:
6370:
6364:
6361:
6358:
6352:
6349:
6346:
6338:
6334:
6325:
6309:
6306:
6303:
6296:If the orbit
6295:
6281:
6275:
6270:
6266:
6259:
6254:
6250:
6246:
6240:
6234:
6231:
6225:
6222:
6219:
6211:
6207:
6200:
6197:
6190:
6189:
6188:
6174:
6171:
6168:
6162:
6158:
6155:
6149:
6146:
6140:
6134:
6126:
6122:
6117:
6113:
6109:
6105:
6100:
6098:
6094:
6078:
6075:
6066:
6060:
6057:
6043:
6039:
6035:
6030:
6028:
6024:
6005:
5999:
5976:
5957:
5954:
5949:
5945:
5941:
5932:
5926:
5920:
5913:
5912:
5911:
5909:
5881:
5880:adjoint group
5852:
5849:
5816:
5813:
5780:
5777:
5744:
5741:
5732:
5730:
5726:
5683:
5680:
5677:
5671:
5665:
5656:
5650:
5647:
5627:
5595:
5527:
5523:
5519:
5513:
5509:
5493:
5490:
5484:
5481:
5478:
5432:
5428:
5424:
5402:
5399:
5395:
5391:
5388:
5385:
5379:
5371:
5367:
5358:
5354:
5350:
5346:
5342:
5332:
5330:
5326:
5322:
5318:
5314:
5310:
5282:
5242:
5233:
5218:
5215:
5194:
5191:
5183:
5164:
5148:
5123:
5120:
5117:
5110:
5109:
5108:
5106:
5074:
5070:
5066:
5060:
5057:
5054:
5047:
5046:
5045:
5043:
5014:
5010:
5006:
4997:
4995:
4985:
4983:
4979:
4975:
4970:
4968:
4964:
4960:
4956:
4952:
4948:
4944:
4925:
4921:
4916:
4912:
4908:
4904:
4900:
4895:
4891:
4887:
4883:
4880:
4872:
4866:
4862:
4853:
4847:
4843:
4839:
4835:
4829:
4825:
4819:
4815:
4810:
4806:
4799:
4798:
4797:
4778:
4772:
4768:
4759:
4753:
4749:
4740:
4736:
4720:
4697:
4692:
4689:
4677:
4671:
4668:
4662:
4656:
4650:
4644:
4638:
4630:
4627:
4622:
4613:
4607:
4604:
4598:
4592:
4584:
4581:
4576:
4570:
4564:
4561:
4555:
4549:
4545:
4541:
4533:
4527:
4523:
4519:
4515:
4510:
4506:
4502:
4498:
4495:
4491:
4485:
4481:
4475:
4471:
4466:
4462:
4455:
4454:
4453:
4451:
4447:
4431:
4408:
4405:
4402:
4393:
4390:
4387:
4381:
4378:
4370:
4367:
4362:
4356:
4353:
4350:
4342:
4339:
4334:
4331:
4328:
4325:
4322:
4319:
4312:
4311:
4310:
4294:
4290:
4286:
4281:
4277:
4271:
4267:
4244:
4240:
4234:
4230:
4221:
4217:
4199:
4195:
4172:
4168:
4159:
4155:
4137:
4133:
4109:
4101:
4095:
4091:
4087:
4079:
4075:
4068:
4061:
4060:
4059:
4045:
4039:
4036:
4033:
4010:
4004:
4001:
3978:
3972:
3969:
3949:
3941:
3931:
3929:
3901:
3882:
3879:
3873:
3870:
3864:
3856:
3853:generated by
3852:
3848:
3844:
3828:
3795:
3785:
3781:
3777:
3763:
3760:
3737:
3734:
3731:
3725:
3721:
3699:
3698:
3697:
3680:
3674:
3671:
3668:
3653:
3648:
3627:
3624:
3619:
3615:
3611:
3605:
3602:
3594:
3590:
3566:
3563:
3552:
3548:
3529:
3526:
3495:
3487:generated by
3461:
3457:
3453:
3445:
3441:
3409:
3405:
3401:
3395:
3392:
3389:
3373:
3358:
3329:
3309:
3308:Ado's theorem
3299:
3285:
3282:
3279:
3271:
3255:
3246:
3243:
3220:
3208:
3202:
3199:
3196:
3183:
3179:
3121:
3118:
3115:
3102:
3099:
3091:
3014:
3005:
2999:
2996:
2993:
2990:
2975:
2972:
2969:
2966:
2950:
2944:
2941:
2917:
2914:
2911:
2908:
2905:
2885:
2884:
2883:
2882:
2881:
2860:
2857:
2853:
2837:
2834:
2831:
2824:
2806:
2802:
2798:
2795:
2789:
2783:
2780:
2777:
2757:
2734:
2728:
2725:
2722:
2702:
2694:
2690:
2685:
2683:
2679:
2675:
2671:
2667:
2663:
2658:
2626:
2622:
2603:
2597:
2594:
2562:
2558:
2555:
2540:
2537:
2534:
2531:
2511:
2505:
2502:
2499:
2491:
2472:
2466:
2463:
2454:
2448:
2445:
2442:
2439:
2431:
2428:
2425:
2421:
2420:
2416:
2415:
2414:
2406:
2391:
2388:
2384:
2378:
2374:
2362:
2358:
2354:
2349:
2345:
2341:
2335:
2329:
2326:
2318:
2299:
2293:
2290:
2282:
2278:
2274:
2258:
2249:
2243:
2240:
2237:
2234:
2226:
2208:
2204:
2198:
2195:
2192:
2188:
2179:
2175:
2171:
2167:
2145:
2139:
2136:
2130:
2127:
2124:
2115:
2109:
2106:
2100:
2092:
2089:(and one for
2076:
2067:
2061:
2058:
2055:
2052:
2045:
2029:
2026:
2006:
2000:
1997:
1994:
1986:
1982:
1977:
1964:
1957:
1954:
1947:
1944:
1941:
1935:
1929:
1926:
1923:
1916:
1913:
1909:
1906:
1900:
1897:
1876:
1873:
1869:
1866:
1843:
1835:
1831:
1824:
1821:
1818:
1815:
1812:
1804:
1800:
1793:
1790:
1787:
1779:
1775:
1771:
1768:
1765:
1760:
1756:
1749:
1746:
1739:
1738:
1737:
1718:
1714:
1707:
1704:
1695:
1689:
1686:
1683:
1678:
1674:
1670:
1648:
1644:
1637:
1634:
1629:
1625:
1616:
1598:
1594:
1590:
1587:
1584:
1579:
1575:
1571:
1568:
1554:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1524:
1505:
1499:
1496:
1492:
1488:
1480:
1476:
1472:
1468:
1464:
1445:
1439:
1436:
1413:
1407:
1404:
1396:
1392:
1369:
1366:
1360:
1354:
1351:
1345:
1339:
1336:
1333:
1327:
1307:
1301:
1298:
1295:
1275:
1269:
1266:
1263:
1255:
1251:
1233:
1227:
1221:
1218:
1209:
1203:
1200:
1196:
1192:
1185:
1184:
1183:
1182:
1180:
1176:
1157:
1154:
1148:
1145:
1142:
1133:
1127:
1124:
1118:
1115:
1107:
1103:
1085:
1082:
1076:
1073:
1070:
1061:
1055:
1052:
1046:
1043:
1036:
1034:
1030:
1026:
1004:
998:
995:
989:
986:
977:
971:
968:
962:
959:
952:
951:
950:
948:
926:
920:
917:
908:
902:
899:
896:
891:
887:
883:
880:
877:
874:
867:
866:
865:
863:
844:
838:
835:
832:
825:
824:
823:
818:Homomorphisms
815:
813:
794:
790:
781:
778:
770:
767:
762:
759:
755:
751:
740:
737:
731:
728:
725:
721:
717:
711:
705:
702:
695:
694:
693:
691:
687:
683:
679:
669:
667:
663:
644:
638:
635:
626:
612:
607:
603:
594:
590:
586:
582:
563:
557:
554:
546:
542:
523:
520:
517:
506:
502:
483:
477:
474:
465:
463:
445:
441:
432:
416:
411:
408:
404:
397:
392:
388:
384:
379:
369:
365:
361:
338:
335:
332:
326:
318:
314:
293:
287:
284:
279:
275:
249:
246:
242:
238:
230:
226:
217:
207:
203:
199:
189:
188:
187:
185:
181:
177:
173:
169:
166:
162:
159:
144:
142:
138:
134:
130:
128:
123:
119:
114:
112:
108:
104:
100:
82:
53:
38:
34:
30:
26:
22:
10924:Lie algebras
10820:Moving frame
10815:Morse theory
10805:Gauge theory
10597:Tensor field
10526:Closed/Exact
10505:Vector field
10473:Distribution
10414:Hypercomplex
10409:Quaternionic
10376:
10146:Vector field
10104:Smooth atlas
10013:
9961:
9958:Lee, John M.
9940:
9911:
9885:
9876:
9853:
9755:
9743:
9731:
9726:, Ch II, § 5
9719:
9707:
9696:
9691:Theorem 2.14
9684:
9672:
9660:
9648:
9636:
9631:Example 3.27
9624:
9619:Theorem 5.20
9612:
9600:
9595:Theorem 5.25
9588:
9576:
9564:
9552:
9540:
9424:
9420:
9414:
9413:
9409:
9408:
9395:
9383:
9371:
9347:Formal group
9270:between the
9116:
9113:Hopf algebra
9075:
9038:
9002:
9000:
8989:can be made
8986:
8982:
8948:
8870:
8868:
8858:
8611:
8599:
8313:
8301:
8036:
8024:
7774:
7714:
7681:
7677:
7673:
7671:
7578:
7548:Killing form
7461:
7425:
7418:
7416:
7397:complex tori
7394:
7261:
7174:
7170:
7137:
7092:
7009:
6965:
6963:
6958:
6954:
6950:
6946:
6944:
6818:
6814:
6810:
6808:
6727:
6725:
6463:
6435:
6433:
6323:
6124:
6120:
6115:
6111:
6107:
6103:
6101:
6096:
6092:
6037:
6033:
6031:
6026:
6022:
5991:
5907:
5733:
5728:
5356:
5352:
5348:
5344:
5338:
5312:
5308:
5280:
5234:
5179:
5104:
5102:
5040:is a (real)
4998:
4991:
4981:
4977:
4973:
4971:
4966:
4962:
4958:
4954:
4950:
4946:
4942:
4940:
4738:
4712:
4449:
4445:
4423:
4219:
4215:
4157:
4153:
4124:
3937:
3927:
3899:
3854:
3846:
3842:
3752:
3651:
3649:
3550:
3439:
3305:
3181:
3089:
3031:
2855:
2692:
2686:
2677:
2673:
2669:
2665:
2661:
2656:
2654:
2624:
2620:
2560:
2556:
2489:
2429:
2417:
2412:
2316:
2276:
2272:
2224:
2177:
2173:
2169:
2165:
2093:) such that
2090:
1984:
1980:
1978:
1858:
1560:
1542:
1538:
1530:
1526:
1522:
1478:
1474:
1466:
1462:
1394:
1390:
1388:
1178:
1105:
1028:
1024:
944:
859:
821:
809:
689:
681:
677:
675:
661:
627:
592:
588:
584:
580:
544:
540:
500:
466:
431:differential
266:
183:
179:
175:
171:
167:
165:vector field
160:
155:
126:
117:
115:
103:circle group
24:
18:
10765:Levi-Civita
10755:Generalized
10727:Connections
10677:Lie algebra
10609:Volume form
10510:Vector flow
10483:Pushforward
10478:Lie bracket
10377:Lie algebra
10342:G-structure
10131:Pushforward
10111:Submanifold
9848:is abelian.
9679:Section 5.7
9655:Section 1.4
9607:Theorem 5.6
9402:Section 3.3
9080:convolution
7766:Root system
7680:is compact
7497:is compact.
7464:is compact.
7428:is compact.
3092:morphisms
2164:and, since
1461:. Also, if
547:are. Thus,
505:Lie bracket
33:Lie algebra
21:mathematics
10929:Lie groups
10913:Categories
10888:Stratifold
10846:Diffeology
10642:Associated
10443:Symplectic
10428:Riemannian
10357:Hyperbolic
10284:Submersion
10192:Hopf–Rinow
10126:Submersion
10121:Smooth map
9903:3540152938
9886:Lie groups
9868:References
6042:unimodular
3268:from the
2695:Lie group
2524:such that
1535:submersion
1256:holds: if
1254:chain rule
111:one-to-one
37:isomorphic
10934:Manifolds
10770:Principal
10745:Ehresmann
10702:Subbundle
10692:Principal
10667:Fibration
10647:Cotangent
10519:Covectors
10372:Lie group
10352:Hermitian
10295:manifolds
10264:Immersion
10259:Foliation
10197:Noether's
10182:Frobenius
10177:De Rham's
10172:Darboux's
10063:Manifolds
10020:EMS Press
9988:808682771
9799:
9770:
9712:Hall 2015
9701:Hall 2015
9689:Hall 2015
9677:Hall 2015
9665:Hall 2015
9653:Hall 2015
9641:Hall 2015
9629:Hall 2015
9617:Hall 2015
9605:Hall 2015
9593:Hall 2015
9581:Hall 2015
9505:
9491:−
9449:−
9438:
9400:Hall 2015
9363:Citations
9218:. By the
9140:
9016:
8815:∣
8788:∈
8661:∣
8634:∈
8550:∣
8523:∈
8423:−
8354:∣
8336:∈
8258:∣
8225:∈
8092:∣
8059:∈
7998:
7992:∣
7962:∈
7836:¯
7827:∣
7797:∈
7698:~
7616:
7577:For each
7511:↪
7478:
7445:~
7353:~
7300:~
7244:→
7233:→
7225:~
7216:→
7198:π
7194:→
7154:~
7109:π
7075:→
7072:Γ
7020:Γ
6989:→
6910:
6890:
6864:
6832:
6741:
6654:
6648:∣
6642:∈
6556:
6527:
6521:∣
6511:∈
6402:→
6374:ρ
6365:
6350:⋅
6307:⋅
6263:→
6235:ρ
6226:
6201:
6172:⋅
6166:↦
6153:→
6135:ρ
6061:
5974:→
5958:
5939:→
5924:→
5853:
5817:
5745:
5725:group law
5651:
5557:→
5517:↦
5488:→
5400:−
5216:π
5195:π
5134:→
5121:π
5064:→
5055:π
5042:Lie group
4867:ϕ
4848:ϕ
4773:ϕ
4754:ϕ
4721:ϕ
4693:⋯
4672:ϕ
4657:ϕ
4639:ϕ
4608:ϕ
4593:ϕ
4565:ϕ
4550:ϕ
4528:ϕ
4452:. Thus,
4432:⋯
4406:⋯
4309:, where
4096:ϕ
4043:→
4005:
3973:
3950:ϕ
3874:
3868:↦
3826:→
3764:
3729:↦
3712:→
3675:
3606:~
3567:~
3530:~
3396:
3359:⊂
3253:→
3247:~
3218:→
3191:Γ
3164:Γ
3128:Γ
3113:→
3103::
3100:ϵ
3039:Γ
2951:≅
2926:Γ
2868:Γ
2850:from the
2693:connected
2598:
2532:ϕ
2509:→
2467:
2461:→
2449:
2440:ϕ
2368:∞
2359:∑
2330:
2294:
2256:→
2244:
2189:⋃
2131:
2110:
2074:→
2062:
2004:→
1948:
1942:∩
1930:
1910:∩
1901:
1825:
1819:⊕
1816:⋯
1813:⊕
1794:
1772:×
1769:⋯
1766:×
1750:
1708:
1702:→
1690:
1641:→
1591:×
1588:⋯
1585:×
1500:
1471:immersion
1440:
1408:
1361:∘
1337:∘
1305:→
1273:→
1222:
1216:→
1204:
1149:
1128:
1119:
1077:
1056:
1047:
999:
963:
921:
915:→
903:
842:→
782:∈
768:∈
752:∣
729:∈
706:
639:
558:
478:
401:→
291:→
133:manifolds
29:Lie group
10866:Orbifold
10861:K-theory
10851:Diffiety
10575:Pullback
10389:Oriented
10367:Kenmotsu
10347:Hadamard
10293:Types of
10242:Geodesic
10067:Glossary
9960:(2012).
9939:(1978),
9515:′
9464:′
9376:Lee 2012
9331:See also
9119:is then
6091:for all
5946:→
5219:′
4713:because
3516:and let
2852:category
2283:), then
1958:′
1917:′
1877:′
1173:and the
1023:for all
676:Suppose
507:; i.e.,
460:between
101:and the
10810:History
10793:Related
10707:Tangent
10685:)
10665:)
10632:Adjoint
10624:Bundles
10602:density
10500:Torsion
10466:Vectors
10458:Tensors
10441:)
10426:)
10422:,
10420:Pseudo−
10399:Poisson
10332:Finsler
10327:Fibered
10322:Contact
10320:)
10312:Complex
10310:)
10279:Section
9427:, then
7093:By the
6817:, then
6730:. Then
6187:. Then
2823:functor
2180:, then
1473:and so
1397:, then
1177:holds:
1027:in Lie(
668:below.
429:is the
10775:Vector
10760:Koszul
10740:Cartan
10735:Affine
10717:Vector
10712:Tensor
10697:Spinor
10687:Normal
10683:Stable
10637:Affine
10541:bundle
10493:bundle
10439:Almost
10362:Kähler
10318:Almost
10308:Almost
10302:Closed
10202:Sard's
10158:(list)
9986:
9976:
9947:
9926:
9900:
7097:, the
6466:, let
6127:. Let
5992:where
4941:Thus,
4214:(with
4125:where
3182:counit
3176:being
1469:is an
267:where
147:Basics
10883:Sheaf
10657:Fiber
10433:Rizza
10404:Prime
10235:Local
10225:Curve
10087:Atlas
5906:. If
5325:SU(2)
5317:SO(3)
5281:every
4424:with
3962:from
3545:be a
2559:: If
2432:: If
1613:be a
1545:is a
1533:is a
1525:. If
945:is a
860:is a
97:(see
31:to a
10750:Form
10652:Dual
10585:flow
10448:Tame
10424:Sub−
10337:Flat
10217:Maps
9984:OCLC
9974:ISBN
9945:ISBN
9924:ISBN
9898:ISBN
9298:and
9001:Let
8610:SO(2
8035:SO(2
7546:The
7417:Let
6945:Let
6326:and
6110:and
6102:Let
6032:Let
4961:and
4949:and
4448:and
4218:and
4187:and
3898:and
3090:unit
2856:left
2672:and
2664:and
2319:and
2196:>
1979:Let
1561:Let
1288:and
1252:The
591:and
467:Let
351:and
124:and
68:and
10672:Jet
9916:doi
9890:doi
9824:as
9796:exp
9767:exp
9502:Lie
9435:Lie
9274:of
9189:in
9137:Lie
9074:on
9037:of
9013:Lie
8981:of
8953:of
8869:If
8688:det
8312:Sp(
8119:det
8039:+1)
7861:det
7777:+1)
7773:SU(
7637:is
7581:in
7550:on
7475:Int
7460:of
7169:of
7040:exp
7008:of
6976:exp
6964:If
6907:Lie
6887:Lie
6861:Lie
6829:Lie
6809:If
6738:Lie
6462:or
6438:of
6223:ker
6198:Lie
6123:in
6095:in
6052:det
6040:is
5955:Int
5882:of
5850:Int
5842:. (
5770:of
5742:Int
5727:on
5596:of
5331:.
5232:.)
4002:Lie
3994:to
3970:Lie
3871:exp
3845:of
3761:Lie
3672:Lie
3549:of
3393:Lie
2623:of
2595:Lie
2464:Lie
2446:Lie
2327:exp
2291:Lie
2241:Lie
2235:exp
2128:exp
2107:exp
2059:Lie
2053:exp
1945:Lie
1927:Lie
1898:Lie
1859:If
1822:Lie
1791:Lie
1747:Lie
1705:Lie
1687:Lie
1497:ker
1437:Lie
1405:Lie
1201:ker
1116:Lie
1074:ker
1053:ker
1044:Lie
996:exp
960:exp
918:Lie
900:Lie
822:If
703:Lie
636:Lie
595:in
555:Lie
475:Lie
464:.
433:of
182:in
170:on
19:In
10915::
10663:Co
10018:,
10012:,
9982:.
9972:.
9964:.
9922:,
9896:,
9327:.
9082:.
8923:dR
7995:tr
7613:ad
7605:,
7392:.
7247:1.
7090:.
6931:.
6806:.
6651:Ad
6553:Ad
6524:ad
6362:im
6099:.
6058:Ad
5950:Ad
5814:ad
5731:.
5648:ad
5628:ad
5479:Ad
5279:,
4631:12
4371:12
3930:.
2684:.
2405:.
1553:)
1219:im
1146:im
1125:im
543:,
186:,
178:,
143:.
113:.
23:,
10681:(
10661:(
10437:(
10418:(
10316:(
10306:(
10069:)
10065:(
10055:e
10048:t
10041:v
9990:.
9918::
9892::
9834:g
9812:)
9807:g
9802:(
9793:=
9788:n
9784:)
9778:g
9773:(
9525:.
9522:)
9519:)
9512:H
9508:(
9499:(
9494:1
9487:)
9483:f
9480:d
9477:(
9474:=
9471:)
9468:)
9461:H
9457:(
9452:1
9445:f
9441:(
9425:H
9421:H
9407:'
9315:)
9312:G
9309:(
9306:A
9284:g
9258:)
9255:G
9252:(
9249:A
9246:=
9243:)
9238:g
9233:(
9230:U
9206:)
9203:G
9200:(
9197:A
9173:)
9170:)
9167:G
9164:(
9161:A
9158:(
9155:P
9152:=
9149:)
9146:G
9143:(
9134:=
9129:g
9117:G
9099:)
9096:G
9093:(
9090:A
9076:G
9058:)
9055:G
9052:(
9049:A
9039:G
9025:)
9022:G
9019:(
9003:G
8987:G
8983:G
8963:g
8934:)
8931:G
8928:(
8919:H
8915:=
8912:)
8908:R
8904:;
8899:g
8894:(
8889:k
8885:H
8871:G
8862:n
8859:D
8843:}
8839:0
8836:=
8833:X
8830:+
8824:T
8819:X
8812:)
8808:C
8804:(
8799:n
8796:2
8792:M
8785:X
8781:{
8777:=
8757:)
8753:C
8749:,
8746:n
8743:2
8740:(
8735:o
8732:s
8707:}
8703:1
8700:=
8697:)
8694:A
8691:(
8685:,
8682:I
8679:=
8676:A
8670:T
8665:A
8658:)
8654:R
8650:(
8645:n
8642:2
8638:M
8631:A
8627:{
8623:=
8614:)
8612:n
8603:n
8600:C
8584:}
8580:0
8577:=
8574:X
8571:J
8568:+
8565:J
8559:T
8554:X
8547:)
8543:C
8539:(
8534:n
8531:2
8527:M
8520:X
8516:{
8512:=
8492:)
8488:C
8484:,
8481:n
8478:(
8473:p
8470:s
8444:]
8438:0
8431:n
8427:I
8414:n
8410:I
8404:0
8398:[
8393:=
8390:J
8386:,
8382:}
8378:J
8375:=
8372:A
8369:J
8363:T
8358:A
8351:)
8348:n
8345:2
8342:(
8339:U
8333:A
8329:{
8325:=
8316:)
8314:n
8305:n
8302:B
8286:}
8282:0
8279:=
8276:X
8273:+
8267:T
8262:X
8255:)
8251:C
8247:(
8242:1
8239:+
8236:n
8233:2
8229:M
8222:X
8218:{
8214:=
8194:)
8190:C
8186:,
8183:1
8180:+
8177:n
8174:2
8171:(
8166:o
8163:s
8138:}
8134:1
8131:=
8128:)
8125:A
8122:(
8116:,
8113:I
8110:=
8107:A
8101:T
8096:A
8089:)
8085:R
8081:(
8076:1
8073:+
8070:n
8067:2
8063:M
8056:A
8052:{
8048:=
8037:n
8028:n
8025:A
8010:}
8007:0
8004:=
8001:X
7989:)
7985:C
7981:(
7976:1
7973:+
7970:n
7966:M
7959:X
7956:{
7953:=
7933:)
7929:C
7925:,
7922:1
7919:+
7916:n
7913:(
7908:l
7905:s
7880:}
7876:1
7873:=
7870:)
7867:A
7864:(
7858:,
7855:I
7852:=
7849:A
7843:T
7833:A
7824:)
7820:C
7816:(
7811:1
7808:+
7805:n
7801:M
7794:A
7790:{
7786:=
7775:n
7736:1
7732:S
7728:=
7725:G
7715:G
7695:G
7678:G
7674:G
7668:.
7654:g
7625:)
7622:X
7619:(
7591:g
7579:X
7560:g
7531:)
7527:R
7523:,
7520:n
7517:(
7514:O
7508:G
7483:g
7462:G
7442:G
7426:G
7419:G
7378:g
7350:G
7325:g
7297:G
7272:g
7241:G
7236:p
7222:G
7213:)
7210:G
7207:(
7202:1
7191:1
7175:G
7171:G
7151:G
7138:G
7124:)
7121:G
7118:(
7113:1
7078:G
7068:/
7062:g
7010:G
6992:G
6984:g
6979::
6966:G
6959:G
6955:G
6951:G
6947:G
6919:)
6916:H
6913:(
6903:/
6899:)
6896:G
6893:(
6884:=
6881:)
6878:H
6874:/
6870:G
6867:(
6841:)
6838:H
6835:(
6819:H
6815:G
6811:H
6794:)
6791:A
6788:(
6782:g
6775:z
6769:=
6766:)
6763:)
6760:A
6757:(
6752:G
6748:Z
6744:(
6728:A
6711:}
6708:A
6700:a
6692:g
6689:a
6686:=
6683:a
6680:g
6672:0
6669:=
6666:a
6663:)
6660:g
6657:(
6645:G
6639:g
6636:{
6633:=
6630:)
6627:A
6624:(
6619:G
6615:Z
6593:}
6590:A
6582:a
6574:0
6571:=
6568:X
6565:)
6562:a
6559:(
6545:0
6542:=
6539:X
6536:)
6533:a
6530:(
6516:g
6508:X
6505:{
6502:=
6499:)
6496:A
6493:(
6487:g
6480:z
6464:G
6448:g
6436:A
6430:.
6418:)
6415:X
6410:x
6406:T
6399:G
6394:e
6390:T
6386::
6383:)
6380:x
6377:(
6371:d
6368:(
6359:=
6356:)
6353:x
6347:G
6344:(
6339:x
6335:T
6324:X
6310:x
6304:G
6282:.
6279:)
6276:X
6271:x
6267:T
6260:G
6255:e
6251:T
6247::
6244:)
6241:x
6238:(
6232:d
6229:(
6220:=
6217:)
6212:x
6208:G
6204:(
6175:x
6169:g
6163:g
6159:,
6156:X
6150:G
6147::
6144:)
6141:x
6138:(
6125:X
6121:x
6116:x
6112:G
6108:X
6104:G
6097:G
6093:g
6079:1
6076:=
6073:)
6070:)
6067:g
6064:(
6055:(
6038:G
6034:G
6027:G
6023:G
6009:)
6006:G
6003:(
6000:Z
5977:0
5971:)
5966:g
5961:(
5942:G
5936:)
5933:G
5930:(
5927:Z
5921:0
5908:G
5892:g
5866:)
5861:g
5856:(
5830:)
5825:g
5820:(
5794:)
5789:g
5784:(
5781:L
5778:G
5758:)
5753:g
5748:(
5729:G
5709:g
5687:]
5684:Y
5681:,
5678:X
5675:[
5672:=
5669:)
5666:Y
5663:(
5660:)
5657:X
5654:(
5606:g
5580:)
5575:g
5570:(
5565:l
5562:g
5552:g
5528:g
5524:c
5520:d
5514:g
5510:,
5507:)
5502:g
5497:(
5494:L
5491:G
5485:G
5482::
5457:g
5433:g
5429:c
5425:d
5403:1
5396:g
5392:h
5389:g
5386:=
5383:)
5380:h
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1965:.
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1955:H
1951:(
1939:)
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1933:(
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1635::
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1101:.
1089:)
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836::
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795:.
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239:=
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223:(
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184:G
180:h
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127:p
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83:n
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54:n
49:R
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