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{{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. ...

...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 ...ition is that every different multiplicative [[associativity|association]] of the <math>n</math> elements is zero. ...

{{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 ...

...algebraic structure]] on a [[vector space]] that describes some properties of objects such as [[Tree (graph theory)|rooted trees]] and [[vector field]]s ...urray Gerstenhaber]] in his work on [[Deformation theory|deformations]] of algebras. ...

...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_''n'', the [[special linear group]] ...

{{Lie groups |Semi-simple}} ...gebra can be seen as the smallest [[real form]] of a corresponding complex Lie algebra, namely the complexification. ...

for any two elements ''a'' and ''b'' of the set. A [[magma (algebra)|magma]] (that is, a set equipped with a binar ...t are neither commutative nor associative, e.g. for the [[multiplication]] of [[sedenion]]s, which are not even [[alternative algebra|alternative]]. ...

...s]], consist of congruence-permutable algebras, but some, like the variety of [[lattice (order)|lattices]], have members that are not congruence-permutab ...> permute.{{r|Bergman|p=122}} A [[variety (universal algebra)|variety]] of algebras <math>\mathcal{V}</math> is referred to as '''congruence-permutable''' when ...

...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]] ...

...flo|year=1977}} and later generalized to arbitrary finite-dimensional Lie algebras by Kontsevich. ...omorphism. It is equivariant with respect to the natural representation of <math>\mathfrak{g}</math> on these spaces, so it restricts to a vector spac ...

...erred to as a '''homotope''' or an [[isotopy of an algebra|'''isotope''']] of the original. ...ve property|associative]]) denoted by juxtaposition. For an element ''a'' of ''A'', define the '''left ''a''-homotope''' <math>A(a)</math> to be the alg ...

...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 ...

...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 ...

...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 ...

{{Short description|Invariant of vertex algebra}} ...tic]] properties of the vertex algebra are logically related to properties of its Zhu algebra or C₂-algebra. ...

...bra]]s, [[flexible algebra]]s (''A''(''BA'')&nbsp;=&nbsp;(''AB'')''A''), [[Lie admissible algebra]]s, and [[power associative]], but are not associative, ...Pierre Tignol]] (1998) "Composition and Triality", chapter 8 in ''The Book of Involutions'', pp 451–511, Colloquium Publications v 44, [[American Mathema ...

...baric algebras''' (also called '''weighted algebra'''). The study of these algebras was started by {{harvs|txt|authorlink=Ivor Malcolm Haddon Etherington|last= ...us types. The laws of inheritance are then encoded as algebraic properties of the algebra. ...

...he '''ternutator''' or '''alternating ternary sum''', it is a special case of the [[n-commutator|''n''-commutator]] for ''n'' = 3, whereas the 2-commutat ==Properties== ...

{{Short description|special kind of topological group - translation from the German Wikipedia}} ...topological group]] that can be written in a certain sense as a limit of [[Lie group]]s.<ref> Hofmann, Morris 2007, p. vii </ref> ...

{{Redirect|Atiyah sequence|the spectral sequence of Atiyah and Hirzebruch|Atiyah&ndash;Hirzebruch spectral sequence}} ...math>''. Explicitly, it is given by the following [[short exact sequence]] of [[vector bundles]] over ''<math> M</math>'': ...