Let us consider an A ∞ algebra with an invariant inner product. The main goal of this paper is to classify the infinitesimal deformations of this A ∞ algebra preserving the inner product and to apply this result to the construction of homology classes on the moduli spaces of algebraic curves. With this aim, we define cyclic cohomology

. We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some methods already known about the computation of the Hochschild and cyclic homologies of CDGAs. In the last section of the paper, we analyze the p-local homology of the iterated bar construction of a CDGA (p prime). 1. Introduction. The description of eÆcient algorithms of homological computation might be considered as a very important question in Homological Algebra, in order to use those processes mainly in the resolution of problems on algebraic topology; but this subject also inuence directly on the development of non so closedareas as Cohomological Physics (in this sense, we nd useful references in [12], [24], [25]) and Secondary Calculus ([14], [27], [28]). Working in the context ...

Abstract. We provide a direct proof that the Hochschild homology of a Z2-graded algebra is Morita invariant. 1.

Abstract. The aim of this note is to show that the generalized supertrace, constructed in another paper of the author, inducing an isomorphism between the Hochschild homology of a superalgebra and that of the superalgebra of square supermatrices of a given type over A, induces, also, an isomorphism between the cyclic homologies of the two superalgebras. 1. Superalgebras and supermatrices Superalgebras (i.e. Z2-graded algebras) are a very important particular case of graded algebras that play an important role in modern mathematics and theoretical physics. They are, particularly, central objects, as basic algebraic machinery in the theory of supermanifolds, which are objects similar to the ordinary manifolds, regarded as ringed spaces, but the sheaf of smooth functions on the manifolds is replaced by a suitable sheaf of superalgebras. Their different kind of homologies can be defined in a way quite similar to the ungraded case, but we have to take into account the grading, which results in extra signs added in suitable places. We are not going to say much about superalgebras, they are described in details,

Extending work of Budzyński and Kondracki, we investigate coverings and gluings of algebras and differential algebras. We describe in detail the gluing of two quantum discs along their classical subspace, giving a C ∗-algebra isomorphic to a certain Podle´s sphere, as well as the gluing of U q 1/2(sl2)-covariant differential calculi on the discs. 1991 MSC: 81R50, 46L87 Keywords: covering; gluing; C*-algebra; differential algebra; quantum disc; quantum sphere; covariance