Double extensions of restricted Lie (super)algebras

Abstract

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

arXiv commentAn improved version; to appear in Arnold Mathematical Journal