Homology and cohomology with coefficients, of an algebra over a quadratic operad

The aim of this paper is to give a definition of groups of homology and cohomology with coefficients, for an algebra over a quadratic operad in characteristic zero. Our work completes the works of Ginzburg and Kapranov (1994) of Kimura and Voronov (1995) and of Fox and Markl (1997). In particular, we emphasize that coefficients for homology and cohomology are radically different. If U-P(A) denotes the enveloping algebra of a P-algebra A, then one can define the homology of A with coefficients in a right module over U-P(A) and the cohomology of A with coefficients in a left module over U-P(A). For classical operads such as those encoding associative, commutative, Lie or Poisson algebras, there is no difference between left module and right module; it is not the case for the operads encoding Leibniz algebras and dual Leibniz algebras. This phenomenon has already been observed by Loday (1995) for Leibniz algebras. At the end of this paper, we study the new case of dual Leibniz algebras and the relations between our theory and Barr-Beck's theory of homology. (C) 1998 Elsevier Science B.V. All rights reserved.