Abstract
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 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 classifies the abelian extensions of dialgebras and Morita invariance of matrices holds for unital dialgebras. 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 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 in [Q] for commutative algebras. Our results follow from a property of Poincaré-Birkhoff-Witt type and from some combinatorial and simplicial properties of the sets of planar binary trees proved in [F4]. Finally, since the faces and degeneracies for unital dialgebras satisfy all the simplicial relations except one, we are lead to study the general properties of the so-called almost simplicial modules.
Keywords
- Spectral Sequence
- Associative Algebra
- Leibniz Algebra
- Abelian Extension
- Unital Algebra
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Rabetti, A. (2001). Dialgebra (co)homology with coefficients. In: Dialgebras and Related Operads. Lecture Notes in Mathematics, vol 1763. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45328-8_3
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DOI: https://doi.org/10.1007/3-540-45328-8_3
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