Double extensions of restricted Lie (super)algebras

TL;DR

This paper investigates the structure of double extensions of restricted Lie (super)algebras with non-degenerate invariant symmetric bilinear forms, showing how such extensions preserve restriction properties and providing new examples, especially in characteristic 3.

Contribution

It demonstrates that double extensions of restricted Lie (super)algebras are themselves restricted and characterizes all restricted NIS-(super)algebras with non-trivial centers as such extensions, introducing new examples.

Findings

Double extensions preserve the restriction property.

Restricted NIS-(super)algebras with non-trivial centers are characterized as double extensions.

New examples of restricted Lie (super)algebras and pre-Lie superalgebras in characteristic 3.

Abstract

A double extension ($\mathscr{D}$ extension) of a Lie (super)algebra $\mathfrak a$ with a non-degenerate invariant symmetric bilinear form $\mathscr{B}$, briefly: a NIS-(super)algebra, is an enlargement of $\mathfrak a$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the best known examples of double extensions of loops algebras. Let $\mathfrak a$ be a restricted Lie (super)algebra with a NIS $\mathscr{B}$. Suppose $\mathfrak a$ has a restricted derivation $\mathscr{D}$ such that $\mathscr{B}$ is $\mathscr{D}$-invariant. We show that the double extension of $\mathfrak a$ constructed by means of $\mathscr{B}$ and $\mathscr{D}$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a $\mathscr{D}$-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of $\mathscr{D}$-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.