##
Leibniz homology of dialgebras of matrices

**Abstract**

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