Double Extensions of Restricted Lie (Super)Algebras

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.