Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2

TL;DR

This paper investigates the structure of nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2, revealing new possibilities for their superdimension and providing examples and classifications.

Contribution

It classifies the possible superdimensions of NISes on simple Lie superalgebras in characteristic 2 and introduces the concept of queerification related to NIS existence.

Findings

Superdimension of NISs can be 0, 1, 0|1, or 1|1 in characteristic 2.

NIS superdimension equals 1|1 if and only if the algebra is a queerification of a simple classically restricted Lie algebra.

Examples of NISs on deformations of simple Lie superalgebras in characteristic 2 are provided.

Abstract

As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2. We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or $0|1$, or $1|1$; it is equal to $1|1$ if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic distinct from 2, see arXiv:1806.05505). We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2. We also recall examples of multiple NISes on simple Lie algebras over non-closed fields.