A Leibniz algebra structure on the second tensor power

R. Kurdiani and T. Pirashvili

Abstract: 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 Brown and Loday (Topology {\bf 26} (1987), 311--335) constructed by Ellis (J. Pure Appl. Algebra {\bf 46} (1987), 111--115 and Glasgow Math. J. {\bf 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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