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- ...algebra]]s. For the topics in the representation theory of Lie groups and Lie algebras, see [[Glossary of representation theory]]. Because of the lack of {{Lie groups}}'''Notations''': ...23 KB (3,462 words) - 21:20, 10 January 2024
- {{Lie groups|Representation}} ...l [[representation theory|representations]] of the complex [[classical Lie groups]] ...20 KB (3,061 words) - 14:36, 20 August 2024
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- ...gebra]], the '''nilradical''' of a [[Lie algebra]] is a nilpotent [[Ideal (Lie algebra)|ideal]], which is as large as possible. ...f a Lie algebra by its nilradical is a [[reductive Lie algebra|reductive]] Lie algebra <math>\mathfrak{g}^{\mathrm{red}}</math>. However, the correspondin ...2 KB (258 words) - 00:02, 2 December 2023
- In geometry, an '''abelian Lie group''' is a [[Lie group]] that is an [[abelian group]]. ...group]] that is a compact group is abelian and a connected compact complex Lie group is a [[complex torus]]; i.e., a quotient of <math>\mathbb{\Complex}^n ...2 KB (247 words) - 14:43, 3 September 2021
- {{Lie groups |Algebras}} ...annihilate ''ξ'' in the [[coadjoint representation]]. The '''index of the Lie algebra''' is ...2 KB (345 words) - 21:07, 25 February 2025
- {{Short description|Concept in Lie algebra mathematics}} {{Lie groups}} ...3 KB (531 words) - 03:00, 27 December 2024
- {{Short description|Restriction on topological groups in mathematics}} ...ly connected space|locally connected]] group with no small subgroup is a [[Lie group]]. (cf. [[Hilbert's fifth problem]].) ...1 KB (157 words) - 02:52, 12 August 2023
- {{Lie groups |Semi-simple}} ...eal [[Lie algebra]] ''g''<sub>0</sub> is called a real form of a [[complex Lie algebra]] ''g'' if ''g'' is the [[complexification]] of ''g''<sub>0</sub>: ...6 KB (943 words) - 15:46, 20 June 2023
- ...ion|Lifts an action of a finite-dimensional Lie algebra on a manifold to a Lie group action}} ...s}} proved it as a global form of an earlier local theorem due to [[Sophus Lie]]. ...3 KB (478 words) - 17:30, 18 August 2024
- ...itted, it is called a '''symmetric Lie algebra'''. An orthogonal symmetric Lie algebra is said to be ''effective'' if <math>\mathfrak{u}</math> intersects The canonical example is the Lie algebra of a [[symmetric space]], <math>s</math> being the differential of ...3 KB (412 words) - 20:12, 12 June 2022
- {{Lie groups |Semi-simple}} ...e.com/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref> Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center. ...5 KB (790 words) - 19:44, 26 January 2024
- {{Lie groups}} ...bra]]s, such as [[SU(N)#Lie algebra|su(''n'')]] and [[special linear group#Lie subgroup|sl(''n'','''R''')]]. ...3 KB (414 words) - 00:41, 19 May 2024
- ...arish-Chandra transform''' is a linear map from functions on a [[reductive Lie group]] to functions on a [[Borel subgroup|parabolic subgroup]]. It was int ...itation | last1=Harish-Chandra | title=Spherical Functions on a Semisimple Lie Group II | jstor=2372772 | publisher=The Johns Hopkins University Press | y ...1 KB (159 words) - 22:25, 10 January 2024
- ...matician and physicist [[Harish-Chandra]], is a representation of a real [[Lie group]], associated to a general representation, with regularity and finite Let ''G'' be a Lie group and ''K'' a compact [[subgroup]] of ''G''. If <math>(\pi,V)</math> i ...2 KB (361 words) - 20:08, 22 March 2021
- ...at is satisfied by all elements of a [[Lie ring]], in the case of an Engel Lie ring, or by all the elements of a [[group (mathematics)|group]], in the cas ...ng <math>L</math>. The Lie ring <math>L</math> is defined to be an n-Engel Lie ring if and only if ...2 KB (277 words) - 02:34, 14 July 2024
- ...[[Lie algebra]] which are invariant under the [[group action|action]] of a Lie group in terms of functions on a [[Cartan subalgebra]]. | ''G'' || complex connected [[semisimple Lie group]] || SL<sub>''n''</sub>, the [[special linear group]] ...2 KB (339 words) - 23:22, 4 February 2025
- *[[Claudio Procesi]] (2007) ''Lie Groups: an approach through invariants and representation'', Springer, {{isbn|9780 [[Category:Lie groups]] ...1 KB (164 words) - 03:02, 13 May 2024
- {{Short description|Lie algebra all of which elements are semisimple}} ...Over an algebraically closed field, every toral Lie algebra is [[abelian Lie algebra|abelian]];<ref name="Hum" /><ref>Proof (from Humphreys): Let <math> ...4 KB (544 words) - 19:56, 5 March 2023
- ...topological group]] that can be written in a certain sense as a limit of [[Lie group]]s.<ref> Hofmann, Morris 2007, p. vii </ref> ...roups is essentially due to the book ''The Lie-Theory of Connected Pro-Lie Groups'' by [[Karl Heinrich Hofmann]] and [[Sidney Morris]], but has since attract ...7 KB (985 words) - 12:46, 20 February 2025
- {{Lie groups |Semi-simple}} ...gebra can be seen as the smallest [[real form]] of a corresponding complex Lie algebra, namely the complexification. ...8 KB (1,179 words) - 20:56, 28 November 2024
- ...>(\mathfrak{g},K)</math>-modules, where <math>\mathfrak{g}</math> is the [[Lie algebra]] of ''G'' and ''K'' is a [[maximal compact subgroup]] of ''G''.<re ...arvnb|Wallach|1988}}</ref> it is a [[vector space]] ''V'' that is both a [[Lie algebra representation]] of <math>\mathfrak{g}</math> and a [[group represe ...4 KB (562 words) - 19:46, 26 January 2024
- ...are used in the theory of [[Lattice (discrete subgroup)|lattices]] in Lie groups, often under the name ''field of definition''. == Fuchsian and Kleinian groups == ...6 KB (1,018 words) - 22:11, 26 March 2024