## 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

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