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- ...ebra]]s. For the topics in the representation theory of Lie groups and Lie algebras, see [[Glossary of representation theory]]. Because of the lack of other op {{Lie groups}}'''Notations''': ...23 KB (3,462 words) - 21:20, 10 January 2024
- ...or <math>\mathfrak{gl}(n)</math> the [[General linear group|general linear Lie algebra]] and <math> I_n </math> the <math> n \times n </math> [[identity ...\in \mathfrak{gl}(n+1) : \text{tr}(x) = 0 \} </math>, the ''special linear Lie algebra''; ...2 KB (278 words) - 14:02, 9 January 2025
- {{Lie groups}} ...r classify) irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the [[theorem of the highest weight]]. ...28 KB (4,267 words) - 05:38, 20 August 2024
- ...3.<ref>{{harvnb|Mubarakzyanov|1963}}</ref> It complements the article on [[Lie algebra]] in the area of [[abstract algebra]]. Let <math>{\mathfrak g}_n</math> be <math>n</math>-dimensional [[Lie algebra]] over the [[field (mathematics)|field]] of [[real number]]s ...8 KB (1,032 words) - 13:42, 24 October 2023
Page text matches
- ...or <math>\mathfrak{gl}(n)</math> the [[General linear group|general linear Lie algebra]] and <math> I_n </math> the <math> n \times n </math> [[identity ...\in \mathfrak{gl}(n+1) : \text{tr}(x) = 0 \} </math>, the ''special linear Lie algebra''; ...2 KB (278 words) - 14:02, 9 January 2025
- {{Short description|Concept in Lie algebra mathematics}} {{Lie groups}} ...3 KB (531 words) - 03:00, 27 December 2024
- {{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
- {{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
- ...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
- {{Short description|Generalization of a Lie algebra}} ...or]] category. Lie conformal algebras are very closely related to [[vertex algebras]] and have many applications in other areas of algebra and integrable syste ...3 KB (449 words) - 19:11, 22 July 2022
- {{Lie groups |Semi-simple}} ...m/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
- ...''' is an [[operad theory|operad]] [[Operad algebra|whose algebras]] are [[Lie algebra]]s. The notion (at least one version) was introduced by {{harvtxt|G ...numbered variables) for variables. Then, <math>\mathcal{Lie} = \{ \mathcal{Lie}(n) \}</math> is an operad.<ref>{{harvnb|Ginzburg|Kapranov|1994|loc=§ 1.3.9 ...2 KB (261 words) - 04:15, 13 May 2024
- ...or space]] ''V''. In other words, a linear Lie algebra is the image of a [[Lie algebra representation]]. Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of <mat ...1 KB (209 words) - 16:08, 15 February 2022
- ...mov, M. T.|title=Unipotent and Nakayama automorphisms of quantum nilpotent algebras|date=1 Nov 2013|class=math.QA|eprint=1311.0278}}</ref> a concept related to ...title=Structure of Algebras|page=22|chapter=Chapt. 2: Ideals and Nilpotent Algebras|orig-year=1939|year=2003|series=Colloquium Publications, Col. 24|publisher= ...3 KB (409 words) - 10:02, 22 April 2021
- ...Lie algebra''' is a [[D-module]] on a curve with a certain structure of [[Lie algebra]]. It is related to an [[E n ring|<math>\mathcal{E}_2</math>-algebr [[Category:Lie algebras]] ...673 bytes (90 words) - 03:14, 13 May 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
- {{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
- ...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
- In [[mathematics]], a '''pre-Lie algebra''' is an [[algebraic structure]] on a [[vector space]] that describ ...urray Gerstenhaber]] in his work on [[Deformation theory|deformations]] of algebras. ...4 KB (686 words) - 01:52, 13 September 2024
- ...'''Mal'tsev Lie algebra''', is a generalization of a rational nilpotent [[Lie algebra]], and Malcev groups are similar. Both were introduced by {{harvtxt ...ding to {{harvtxt|Papadima|Suciu|2004}} a Malcev Lie algebra is a rational Lie algebra <math> L </math> together with a complete, descending <math> {\math ...3 KB (442 words) - 12:47, 4 October 2021
- ...Kac|1990|loc=§ 3.6.}}</ref> For example, the [[adjoint representation of a Lie algebra|adjoint representation]] of a Kac–Moody algebra is [[integrable]].< ...irst=Victor|last=Kac|author-link=Victor Kac|title=Infinite dimensional Lie algebras|edition= 3rd |publisher= [[Cambridge University Press]] |year=1990|isbn=0-5 ...1 KB (145 words) - 04:03, 13 May 2024
- ...square is the identity.<ref name=Schafer>{{cite news|title=On Structurable algebras|author=R.D. Schafer|journal=[[Journal of Algebra]]|year=1985|volume=92|page ...ructurable algebra'' if:<ref name=Garibaldi>{{cite news|title=Structurable Algebras and Groups of Type E_6 and E_7|author=Skip Garibaldi|author-link=Skip Garib ...2 KB (356 words) - 19:07, 29 December 2020
- ...they satisfy the flexible identity.<ref>Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”, [[American Journal of Mathematics]] Besides [[associative algebra]]s, the following classes of nonassociative algebras are flexible: ...3 KB (339 words) - 17:37, 21 February 2025
- {{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