2018/10/07 by Said Benayadi, Benayadi, Said, Sofiane Bouarroudj +3
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1810.03086
An improved version; to appear in Arnold Mathematical Journal
arxiv created 2020/07/25 · arxiv updated 2020/07/28
A double extension (\mathscrD extension) of a Lie (super)algebra \mathfrak a with a non-degenerate invariant symmetric bilinear form \mathscrB, 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 \mathscrB. Suppose \mathfrak a has a restricted derivation \mathscrD such that \mathscrB is \mathscrD-invariant. We show that the double extension of \mathfrak a constructed by means of \mathscrB and \mathscrD is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a \mathscrD-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of \mathscrD-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.