Leibniz homology of dialgebras of matrices

A dialgebra $D$ is a vector space with two associative operations $\lp$, $\rp$ satisfying three more relations. By setting $[x,y]:= x \lp y - y \rp x$, any dialgebra gives rise to a Leibniz algebra. Here we compute the Leibniz homology of the dialgebra of matrices $\gl(D)$ with entries in a given dialgebra $D$. We show that $HL(\gl(D))$ is isomorphic to the tensor module over $HHS(D)$, which is a variation of the natural dialgebra homology $HHY(D)$.

Contents

1 - Variation $HHS$ of dialgebra homology
2 - Leibniz homology of matrices of dialgebras
-- Main theorem
-- The complex $L_{*}(D)$
-- The complex of primitives, $P_{*}(D)$
-- Isomorphism between $P_{*}(D)$ and $CS_{*}(D)$
3 - Comparison between the dialgebra homology $HY$ and its variation $HS$
References