Structure of Degenerate Block Algebras

Abstract

Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map $\phi$ : A $\times$ A $\rightarrow$ F, we define a Lie algebra ${\cal L}$ = ${\cal L}$ (A, $\phi$) over F with basis {e_x | x $\in$ A/{0}} and Lie product [e_x,e_y] = $\phi$(x,y)e_x+y. We show that ${\cal L}$ is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der ${\cal L}$ of ${\cal L}$ is a complete Lie algebra. We describe the double extension D(${\cal L}$, T) of ${\cal L}$ by T, where T is spanned by the locally finite derivations of ${\cal L}$, and determine the second cohomology group H²(D(${\cal L}$, T),F) using anti-derivations related to the form on D(${\cal L}$, T). Finally, we compute the second Leibniz cohomology groups HL²(${\cal L}$, F) and HL²(D(${\cal L}$, T), F).