Double extensions of restricted Lie (super)algebras

Abstract

A double extension (D-extension) of a Lie (super)algebra a with a nondegenerate invariant supersymmetric bilinear form B, briefly a NIS-(super)algebra, is an enlargement of a by means of a central extension and a derivation; the affine Kac-Moody algebras are the best known examples.Let a be a restricted Lie (super)algebra with a NIS B. Suppose a has a restricted derivation D such that B is D-invariant. We show that the double extension of a caries a p-structure constructed by means of B and D. We show that, the other way round, any restricted NIS-(super)algebra can be obtained as a D-extension of another restricted NIS-(super)algebra of codimension 2 provided the center is non-trivial together with an extra condition on the central element.We give examples of D-extensions of restricted Lie (super)algebras in small characteristic related to Manin triples for the Heisenberg algebra, Lie superalgebras of rank 2, and selected examples for vectorial Lie superalgebras. Almost all of them are new.