Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras

TL;DR

This paper reviews non-degenerate invariant bilinear forms on various simple Lie (super)algebras, examining their existence, deformation behavior, and relations to affine Kac--Moody superalgebras, especially over fields of positive characteristic.

Contribution

It provides a comprehensive classification of NIS on simple Lie (super)algebras and analyzes their deformation properties and connections to extended algebras.

Findings

Most deformed Lie (super)algebras with NIS retain the same Gram matrix.

Identifies conditions for the existence of NIS on various classes of Lie (super)algebras.

Establishes links between NIS and doubly extended Lie (super)algebras like affine Kac--Moody superalgebras.

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.