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{{Short description|Concept in Lie algebra mathematics}} {{Lie groups}} ...

{{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. ...

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

...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 ...

...re used to present the exceptional symmetric domains:<ref>Guy Roos (2005) "Exceptional Symmetric Domains", page 158 in ''Symmetries in Complex Analysis'', editors ...e to describe the exceptional symmetric domain of dimension 27. The second exceptional symmetric domain (of complex dimension 16) lives in the space <math>M_{2,1} ...

...in other respects they can give rise to consequential phenomena, such as [[exceptional object]]s.{{sfn|ps=|Wilson|2009|loc=[http://www.maths.qmul.ac.uk/~raw/fsgs_ The exceptional isomorphisms between the series of [[finite simple group]]s mostly involve ...

...baric algebras''' (also called '''weighted algebra'''). The study of these algebras was started by {{harvs|txt|authorlink=Ivor Malcolm Haddon Etherington|last= In applications to genetics, these algebras often have a basis corresponding to the genetically different [[gamete]]s, ...

...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''': ...

...t examples of finite-dimensional Nichols algebras are the [[Borel subgroup#Lie algebra|Borel parts]] <math>u_q(\mathfrak{g})^+</math> of the Frobenius–Lus ...is generated by the support of <math>V</math>. For more details on Nichols algebras see [[Nichols algebra]]. ...

{{about|the cellular algebras of Graham and Lehrer|the cellular algebras of Weisfeiler and Lehman|coherent algebra}} ...r of Graham and Lehrer.<ref name="grahamlehrer">{{citation|title= Cellular algebras|first1 = J.J|last1= Graham|first2= G.I.|last2 = Lehrer|journal= Inventiones ...

..., the theory of quadratic Jordan algebras reduces to that of linear Jordan algebras. ...al''' if it is isomorphic to a subalgebra of such an algebra, otherwise '''exceptional'''.<ref name=Rac2/> ...

{{Lie groups|Homogeneous spaces}} ...in the [[complexification (Lie group)|complexification]] of the compact [[Lie group]], a [[reductive algebraic group]]. ...

{{Short description|Non-associative algebras with positive-definite quadratic form}} ...ere are no other possibilities. Such algebras, sometimes called '''Hurwitz algebras''', are examples of [[composition algebra]]s. ...

{{Lie groups|Representation}} ...ional [[representation theory|representations]] of the complex [[classical Lie groups]] ...

...ormal field theories]], whose symmetry is described by an abelian [[affine Lie algebra]]. === Abelian affine Lie algebra === ...

...i |date=1995 |title=Two-Dimensional Conformal Geometry and Vertex Operator Algebras |url=https://link.springer.com/book/10.1007/978-1-4612-4276-5 |journal=Spri ...choices of <math>\mathfrak{g}</math>, <math>k</math> not lying is certain exceptional families, the category <math>\mathcal{C}({{\mathfrak g}},k)</math> is modul ...

...cated structures called [[Jordan triple system]]s, which generalize Jordan algebras without identity.<ref>This article uses as its main sources {{harvtxt|Jorda ...the interior of its closure, since any interior point in the closure must lie in the interior of some polytope in the original cone. A convex cone is sai ...

...(The name heterotic SO(32) is slightly inaccurate since among the SO(32) [[Lie group]]s, string theory singles out a quotient Spin(32)/Z₂ that ...onstandard signature of spacetime, superstrings, and the split composition algebras |date=1990 |last1=Foot |first1=R. |last2=Joshi |first2=G. C. |journal=Lette ...

...that only Jordan pairs associated with period two automorphisms of Jordan algebras are required. ==Quadratic Jordan algebras== ...

...annian]] for the [[symplectic group]] Sp(4,''k''). (This is related to the exceptional isomorphism of [[linear algebraic group]]s between SO(5,''k'') and <math>\o ...ivalently, of lines in <math>\mathbf{P}^3</math>). (This is related to the exceptional isomorphism of linear algebraic groups between SO(6,''k'') and <math>\opera ...