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- {{Lie groups}} ...esentations 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
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- ...e case of an [[Engel group]]. The Engel identity is the defining condition of an [[Engel group]]. ...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 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
- {{Short description|Lie algebra all of which elements are semisimple}} ...\lambda = 0</math>, we have that <math>-\lambda y</math> is an eigenvector of <math>\operatorname{ad}_{\mathfrak{h}}(y)</math> with eigenvalue zero, a co ...4 KB (544 words) - 19:56, 5 March 2023
- ...ch 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
- ...d compact Lie group|irreducible representations]] of a connected compact [[Lie group]] <math>K</math>.<ref name="12.6">{{harvnb|Hall|2015}} Theorem 12.6</ ...n the two results is in the precise notion of "integral" in the definition of a dominant integral element. If <math>K</math> is simply connected, this di ...8 KB (1,200 words) - 02:04, 30 January 2025
- ...morphism]]s are [[Lie_algebra_representation#Homomorphisms|homomorphisms]] of representations. ...h>\mathfrak{g}</math> is a (usually [[Complex number|complex]]) semisimple Lie algebra with a [[Cartan subalgebra]] ...3 KB (517 words) - 22:31, 24 February 2021
- {{Short description|Formula in representation theory}} ...imple complex [[Lie algebra]] in a [[tensor product]] of two [[irreducible representation]]s. ...2 KB (255 words) - 01:39, 29 April 2024
- ...''' of a [[Lie algebra]] <math>\mathfrak{g}</math> is a maximal [[solvable Lie algebra|solvable]] subalgebra.<ref>{{harvnb|Humphreys|loc=Ch XVI, § 3.}}</r ...bra of a [[complex Lie group]], then a Borel subalgebra is the Lie algebra of a [[Borel subgroup]]. ...4 KB (555 words) - 03:12, 13 May 2024
- {{Short description|Type of automorphism}} ...denoted <math>\text{Aut}(\mathfrak{g})</math>, the [[automorphism group]] of <math>\mathfrak{g}</math>. ...5 KB (769 words) - 20:32, 14 October 2023
- ...H'' of a [[finite group]] ''G''. It is a special case of a [[Hecke algebra of a locally compact group]]. ...]] zero, ''G'' a finite [[group (mathematics)|group]] and ''H'' a subgroup of ''G''. Let <math>F[G]</math> denote the ...2 KB (362 words) - 02:15, 15 May 2024
- ...system]] <math>\Phi</math>, there exists a finite-dimensional [[semisimple Lie algebra]] whose root system is the given <math>\Phi</math>. ...ase <math>\{ \alpha_1, \dots, \alpha_n \}</math> of <math>\Phi</math>, the Lie algebra <math>\mathfrak g</math> defined by (1) <math>3n</math> generators ...7 KB (1,149 words) - 12:43, 15 November 2024
- ...ple Lie algebra]], or a [[Weyl module]] of a [[reductive algebraic group]] of positive characteristic. Jantzen filtrations were introduced by {{harvs|txt If ''M''(λ) is a Verma module of a semisimple Lie algebra with highest weight λ, then the Janzen filtration is a decreasing f ...3 KB (346 words) - 07:43, 23 June 2022
- ...factor for a [[von Neumann algebra]]: the representation <math>\pi </math> of G is isotypical iff <math>\pi(G)^{''} </math> is a factor. This term more generally used in the context of [[semisimple module]]s. ...3 KB (371 words) - 18:50, 1 January 2024
- ...| url=https://archive.org/details/quantumgroups0000kass }}</ref> Given a [[Lie algebra]] <math>\mathfrak{g}</math>, the quantum enveloping algebra is typi ...applications, studying the <math>q \to 0</math> limit led to the discovery of [[crystal base]]s. ...3 KB (395 words) - 06:29, 13 May 2024
- ...tic zero. Historically, they are regarded as leading to the discovery of [[Lie algebra cohomology]].<ref>{{harvnb|Jacobson|1979|loc=p. 93}}</ref> ...r cohomology, respectively, but there are similar statements pertaining to Lie algebra cohomology in arbitrary orders which are also attributed to Whitehe ...6 KB (1,010 words) - 17:28, 28 February 2022
- ...(discrete subgroup)|lattices]] in Lie groups, often under the name ''field of definition''. ...field generated over <math>\mathbb Q</math> by the traces of all elements of <math>\Gamma</math> (see for example in {{harvtxt|Maclachlan|Reid|2003}}). ...6 KB (1,018 words) - 22:11, 26 March 2024
- ...n_for = Writing ''Introduction to Lie Algebras and Representation Theory''<br/>Receiving the [[Lester R. Ford Award]] in 1976 ...g/publications/maa-reviews/introduction-to-lie-algebras-and-representation-theory}}</ref> and ''Reflection Groups and Coxeter Groups''.<ref>{{cite book|publi ...7 KB (951 words) - 14:50, 23 September 2024
- ...e]] [[unitary representation]]s of some classes of [[Lie group]]s by means of the [[orbit method]]<ref>{{cite journal | title = Rationally varying polarizing subalgebras in nilpotent Lie algebras ...6 KB (919 words) - 16:20, 14 September 2024
- ...' is a [[morphism]] that combines [[symplectic geometry]] and [[resolution of singularities]].<ref name="Kamnitzer-2022">{{cite arxiv |last=Kamnitzer |fi ...'Hamiltonian''' if it possesses [[Hamiltonian action|Hamiltonian actions]] of a [[torus]] <math>T</math> on both <math>X</math> and <math>Y</math>. In th ...3 KB (395 words) - 09:10, 21 February 2025
- ...this structure, because the Lie algebra of a [[group scheme]] over a field of characteristic ''p'' is restricted. ...sentation#Adjoint representation of a Lie algebra|adjoint representation]] of <math>\mathfrak{g}</math> is defined by <math>(\text{ad }X)(Y)=[X,Y]</math> ...14 KB (2,230 words) - 01:53, 30 December 2023