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Non-degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras

Authors
  • Bouarroudj, Sofiane1
  • Krutov, Andrey2, 3
  • Leites, Dimitry1, 4
  • Shchepochkina, Irina3
  • 1 New York University Abu Dhabi, Division of Science and Mathematics, Abu Dhabi, United Arab Emirates , Abu Dhabi (United Arab Emirates)
  • 2 Polish Academy of Sciences, Institute of Mathematics, ul. Śniadeckich 8, Warszawa, 00-656, Poland , Warszawa (Poland)
  • 3 Independent University of Moscow, Bolshoj Vlasievsky per, dom 11, Moscow, RU-119 002, Russia , Moscow (Russia)
  • 4 Stockholm University, Department of Mathematics, Stockholm, SE-106 91, Sweden , Stockholm (Sweden)
Type
Published Article
Journal
Algebras and Representation Theory
Publisher
Springer Netherlands
Publication Date
Jun 27, 2018
Volume
21
Issue
5
Pages
897–941
Identifiers
DOI: 10.1007/s10468-018-9802-8
Source
Springer Nature
Keywords
License
Yellow

Abstract

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.

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