Dialgebra (co)homology with Coeecients

Dialgebras are a generalization of associative algebras which gives rise to Leibniz algebras instead of Lie algebras. In this paper we deene the dialgebra (co)homology with coeecients, recovering, for constant coeecients, the natural bar homology of dialgebras introduced by J.-L. Loday in L6] and denoted by HY. We show that the homology HY has the main expected properties: it is a derived functor, HY 2 classiies the abelian extensions of dialge-bras and Morita invariance of matrices holds for bar-unital dialgebras (the best analogue to unital associative algebras). For associative algebras, we compare Hochschild and dialgebra homology, and extend the isomorphism proved in F2] for unital algebras to the case of H-unital algebras. A feature of the theory HY is that the categories of coeecients for homology and coho-mology are diierent. This leads us to introduce the universal enveloping algebra of dialgebras and the corresponding cotangent complex, analogue to that deened by D. Quillen for com-mutative algebras. Our results follow from a property of Poincar e-Birkhoo-Witt type and from some combinatorial and simplicial properties of the sets of planar binary trees proved in F4]. Finally, remarking that for bar-unital dialgebras the faces and degeneracies satisfy all the simplicial relations except one, leads us to study the general properties of the so-called almost simplicial modules.