Dialgebras are a generalization of associative algebras which gives rise to Leibniz algebras instead of Lie algebras. In this paper we define the dialgebra (co)homology with coefficients, recovering, for constant coefficients, the natural bar homology of dialgebras introduced by J.-L. Loday in [Loday 41] and denoted by $HY_{*}$. We show that the homology $HY_{*}$ has the main expected properties: it is a derived functor, $HY^{2}$ classifies the abelian extensions of dialgebras 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 [Frabetti 2] for unital algebras to the case of H-unital algebras.

A feature of the theory $HY$ is that the categories of coefficients for homology and cohomology are different. This leads us to introduce the universal enveloping algebra of dialgebras and the corresponding cotangent complex, analogue to that defined by D. Quillen for commutative algebras. Our results follow from a property of Poincar\'e-Birkhoff-Witt type and from some combinatorial and simplicial properties of the sets of planar binary trees proved in [Frabetti 4]. 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-calledalmost simplicial modules.

ContentsIntroduction

1 - Dialgebras

2 - Dialgebra cohomology and representations

3 - Universal enveloping algebra of dialgebras

4 - Dialgebra homology and corepresentations

5 - Dialgebra cohomology as derived functor

6 - Homology of bar-unital dialgebras

7 - HY-unital dialgebras

References