Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

TL;DR

This paper classifies finite dimensional modular Lie superalgebras with indecomposable Cartan matrices, discovering new simple types, clarifying foundational notions, and exploring phenomena unique to characteristic 2.

Contribution

It provides a comprehensive classification of these superalgebras, introduces new concepts and clarifications, and reports eleven new exceptional simple modular Lie superalgebras.

Findings

Eleven new exceptional simple modular Lie superalgebras discovered.

In characteristic 2, all superalgebras with Cartan matrix derive from simple Lie algebras.

Existence of simple Lie superalgebras with solvable even parts.

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.