Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras
| Autor: | Irina Shchepochkina, Andrey Krutov, Sofiane Bouarroudj, Dimitry Leites |
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| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
General Mathematics 010102 general mathematics Primary 17B50 Secondary 17B20 FOS: Physical sciences Lie group Lie superalgebra Mathematical Physics (math-ph) Mathematics - Rings and Algebras Bilinear form Killing form 01 natural sciences Rings and Algebras (math.RA) 0103 physical sciences Lie algebra FOS: Mathematics Cartan matrix 010307 mathematical physics Representation Theory (math.RT) 0101 mathematics Algebraically closed field Invariant (mathematics) Mathematics - Representation Theory Mathematical Physics Mathematics |
| Popis: | We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac--Moody (super)algebras are the most known examples. 42 pages. Definitions of certain Lie (super)algebras follow previous works of some of the authors |
| Databáze: | OpenAIRE |
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