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A direct sum of simple Lie algebras is called a [[semisimple Lie algebra]]. == Complex simple Lie algebras == ...

...orphism is unchanged when a [[Reidemeister move]] is applied to the tensor diagram and thus they present a representation of the [[braid group]]. ...roup) possess an [[Root system|arithmetic root system]], multiple [[Dynkin diagram]]s and a [[Poincaré–Birkhoff–Witt theorem|PBW-basis]] made up of braided co ...

...on the [[Jones polynomial]] of [[alternating tangles]]. Alternating planar algebras provide an appropriate algebraic framework for other [[knot invariants]] in ...on among input discs of the diagram and its arcs, namely, the union of the diagram arcs and the boundary of the internal holes is a connected set. ...

...m/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref> Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center. ...s by diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields. ...

...ex [[Lie group]]s. Real forms of complex [[semisimple Lie group]]s and Lie algebras have been completely classified by [[Élie Cartan]]. Using the [[Lie correspondence|Lie correspondence between Lie groups and Lie algebras]], the notion of a real form can be defined for Lie groups. In the case of ...

...of [[vertex algebra]]s, which are based on [[formal power series]]. Chiral algebras on curves are essentially [[vertex algebra#vertex operator algebra|conforma ...k |last1=Ben-Zvi |first1=David |last2=Frenkel |first2=Edward |title=Vertex algebras and algebraic curves |date=2004 |publisher=American Mathematical Society |l ...

=== Diagram automorphisms === ...assification of Lie algebras.<ref>{{harvnb|Humphreys|1972}}</ref> The only algebras with non-trivial outer automorphism group are therefore <math>A_n (n \geq 2 ...

* Compact Lie algebras are [[reductive Lie algebra|reductive]];<ref>{{Harv|Knapp|2002|loc=Proposit ...ex semisimple Lie algebras by viewing them as the complexifications of Lie algebras of compact groups;<ref>{{harvnb|Hall|2015}} Chapter 7</ref> the existence o ...

...heory)|real forms]] of the complex Lie algebra corresponding to the Dynkin diagram. ...ductive [[algebraic group]] over a field is a generalization of the Satake diagram to arbitrary fields, introduced by {{harvs|txt|last=Jacques Tits|authorlink ...

...s the [[quiver (mathematics)|quivers]] of finite type in terms of [[Dynkin diagram]]s. ...the arrows are ignored) is one of the [[ADE classification|ADE]] [[Dynkin diagram]]s: <math>A_n</math>, <math>D_n</math>, <math>E_6</math>, <math>E_7</math>, ...

...ory <math>\mathcal{C}</math> such that the following diagram [[Commutative diagram|commutes]]<blockquote><math>\begin{matrix} ...ath>g: B \to B'</math> in <math>\mathcal{C}</math> such that the following diagram commutes<blockquote><math>\begin{matrix} ...

...liott's classification of [[approximately finite-dimensional C*-algebra|AF-algebras]] and the theory of [[subfactor]]s. Subsequently [[Anatoly Vershik]] assoc A Bratteli diagram is given by the following objects: ...

{{Short description|Graphical representation of supersymmetric algebras}} ...' are a graphical representation of [[Supersymmetry algebra|supersymmetric algebras]].<ref name=":0">{{Cite journal | last1 = Faux | first1 = M. | last2 = Gate ...

{{Short description|2-parameter family of algebras with the Hecke algebra of the symmetric group as a quotient}} ...s a deformation of the [[Brauer algebra]] in much the same way that Hecke algebras are deformations of the [[group ring|group algebra]] of the symmetric group ...

...groups]] like <math>SU_q(3)</math> can be realised as the <math>C^*</math>-algebras of <math>k</math>-graphs.<ref>{{citation |first1=O.|last1=Giselsson |title= ...t1=A.|last1=Sims |title=''Lecture notes on higher-rank graphs and their C*-algebras''|url=https://www.aidansims.com/files/k-graphNotes2010.pdf}}</ref> In parti ...

...1=Costello |first1=Kevin |last2=Gwilliam |first2=Owen |title=Factorization algebras in quantum field theory, Volume 1 |date=2017 |location=Cambridge |isbn=9781 === Prefactorization algebras === ...

...eft ideals in Φ(''A'') from the structure theory of ideals for von Neumann algebras, which is relatively much more simple. If ''K'' is a norm-closed left ideal ...ach ''a'' in ''A''. This can be conveniently captured in the [[commutative diagram]] below : ...

'''Cluster algebras''' are a class of [[commutative ring]]s introduced by {{harvs|txt|last=Fomi ...luster algebras of finite type can be classified in terms of the [[Dynkin diagram]]s of finite-dimensional [[simple Lie algebra]]s. ...

...tteli]]. Later, [[George A. Elliott]] gave a complete classification of AF algebras using the ''K''₀ functor whose range consists of [[ordered group ...follows from what is known as ''the intertwining argument''. For unital AF algebras, both existence and uniqueness follow from the fact the Murray-von Neumann ...

...ts|1972}} (except that both these papers accidentally omitted the [[Dynkin diagram]] {{Dynkin|node|4b|nodeg|4a|node}}). ...educed simple roots α (with 2α a root) are colored green. The first Dynkin diagram in a series sometimes does not follow the same rule as the others. ...