Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Abstract

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described.

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• Some new simple modular Lie superalgebras
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Cited by in corpus (19)

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• A series of algebras generalizing the octonions and Hurwitz-Radon identity
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• Restricted simple Lie (super)algebras in characteristic $3$
• From octonions to composition superalgebras via tensor categories
• Deformations of Symmetric Simple Modular Lie (Super)Algebras
• Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2
• From the Albert algebra to Kac's ten-dimensional Jordan superalgebra via tensor categories in characteristic 5
• Inverses of Cartan matrices of Lie algebras and Lie superalgebras
• The roots of exceptional modular Lie superalgebras with Cartan matrix
• Duflo-Serganova homology for exceptional modular Lie superalgebras with Cartan matrix
• Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2
• Classification of finite-dimensional Lie superalgebras whose even part is a three-dimensional simple Lie algebra over a field of characteristic not two or three
• Classification of simple algebras in Deligne category $Rep(S_t)$
• Associative forms and second cohomologies of Lie superalgebras $HO$ and $KO$
• Semisimplifying Lie algebras of $J$-ternary algebras in characteristic $3$
• Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$
• Semisimplification of contragredient Lie algebras