Double extensions of restricted Lie algebras

Said Benayadi, Sofiane Bouarroudj

Published 2018-10-07Version 1

A double extension ($D$-extension) of a Lie algebra ${\mathfrak a}$ with a non-degenerate invariant symmetric bilinear form $B_{\mathfrak a}$, briefly a NIS Lie algebra, is an enlargement of ${\mathfrak a}$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the most known examples. Let ${\mathfrak a}$ be a restricted Lie algebra equipped with a NIS $B_{\mathfrak a}$. Suppose $\mathfrak a$ has a restricted derivation $D$ such that $B_{\mathfrak a}$ is $D$-invariant. We show that the double extension of $\mathfrak a$ caries a $p$-mapping constructed by means of $B_{\mathfrak a}$ and $D$. We show that, the other way round, any restricted NIS Lie algebra can be obtained as a $D$-extension of another restricted NIS Lie algebra of codimension 2 provided that the center is not trivial together with an extra condition pertaining to the central element. We give examples of $D$-extensions of restricted Lie algebras in small characteristic related with Manin triples for the Heisenberg algebra, and with the Hamiltonian and the Jacobson-Witt algebras. These $D$-extensions are classified up to an isometry; some of them are new.