A Leibniz Algebra Structure on the Second Tensor Product

R. Kurdiani ; T. Pirashvili

Journal of Lie theory, Tome 12 (2002) no. 2, pp. 583-596.

Résumé

\def\g{{\mathfrak g}} \newcommand{\tp}{\otimes} \newcommand{\utp}{\underline\otimes} \newcommand{\bt}{\boxtimes}For any Lie algebra $\g$, the bracket $$ [x\tp y,a\tp b]:=[x,[a,b]]\tp y+x\tp [y,[a,b]] $$ defines a Leibniz algebra structure on the vector space $\g \tp \g$. We let $\g\utp\g$ be the maximal Lie algebra quotient of $\g\tp \g$. We prove that this particular Lie algebra is an abelian extension of the Lie algebra version of the nonabelian tensor product $\g \bt \g $ of R. Brown and J.-L. Loday [Topology 26 (1987) 311--335] constructed by G. J. Ellis [J. Pure Appl. Algebra 46 (1987) 111--115; Glasgow Math. J. 33 (1991) 101--120]. We compute this abelian extension and Leibniz homology of $\g\tp \g$ in the case, when $\g$ is a finite dimensional semi-simple Lie algebra over a field of characteristic zero.