On the cohomology and deformations of differential graded algebras

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In this paper we work out the deformation theory for differential graded algebras (dga's) and for differential graded Hopf algebras (dgha's). The constructions generalize the theory of deformations of algebras developed in late sixties by Gerstenhaber and of Hopf algebras, introduced more recently by Gerstenhaber and Schack and the authors. Namely, we introduce a cohomology theory for dga's and for dgha's, “controlling” their deformations. Our main example of a dga will be the de Rham algebra Ω of a smooth algebraic variety. We prove that H •(Ω, M) = H •(M) for any symmetric dg module M over Ω. From this result we deduce that the deformation cohomology of the de Rham algebra of a Lie group coincides with cohomology of its classifying space. We introduce the notion of a Poisson-de Rham Lie group — this is just a usual Poisson Lie group with a graded Poisson bracket on its de Rham algebra extending the Poisson bracket on functions. We prove that for any simple Lie group G the standard Poisson structure cannot be extended to a Poisson-de Rham structure. Hence, there are no deformations of the de Rham algebra of G extending the Drinfeld-Jimbo deformation.