Journal of Lie Theory 12 (2002), No. 2, 583--596
Copyright Heldermann Verlag 2002
A Leibniz Algebra Structure on the Second Tensor Product
R. Kurdiani
A. Razmadze Math. Institute, Georgian Academy of Sciences, Aleksidze Str. 1, Tbilisi 380093, Georgia
T. Pirashvili
A. Razmadze Math. Institute, Georgian Academy of Sciences, Aleksidze Str. 1, Tbilisi 380093, Georgia
[Abstract-pdf]
\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.
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