We consider cohomology of diagrams of algebras by Beck’s approach, using comonads. We then apply this theory to computing the cohomology of Ψ-rings. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of an algebra to the cohomology of the underlying algebra.

We consider a finite group acting on a vector space and the corresponding skew group algebra generated by the group and the symmetric algebra of the space. This skew group algebra illuminates the resulting orbifold and serves as a replacement for the ring of invariant polynomials, especially in the eyes of cohomology. One analyzes the Hochschild cohomology of the skew group algebra using isomor...

For a smooth manifold equipped with a compact Lie group action, we construct an equivariant cohomology theory which takes values in a vertex algebra, and contains the classical equivariant cohomology as a subalgebra. The main idea is to synthesize the algebraic approach to the classical equivariant cohomology theory due to H. Cartan and Guillemin-Sternberg, with the chiral de Rham algebra of Ma...

For a smooth manifold equipped with a compact Lie group action, we construct an equivariant cohomology theory which takes values in a vertex algebra, and contains the classical equivariant cohomology as a subalgebra. The main idea is to synthesize the algebraic approach to the classical equivariant cohomology theory due to H. Cartan, with the chiral de Rham algebra of Malikov-Schechtman-Vaintro...

For a smooth manifold equipped with a compact Lie group action, we construct an equivariant cohomology theory which takes values in a vertex algebra, and contains the classical equivariant cohomology as a subalgebra. The main idea is to synthesize the algebraic approach to the classical equivariant cohomology theory due to H. Cartan and Guillemin-Sternberg, with the chiral de Rham algebra of Ma...

Let LX be the space of free loops on a simply connected manifoldX. When the real cohomology of X is a tensor product of algebras generated by a single element, we determine the algebra structure of the real cohomology of LX by using the cyclic bar complex of the de Rham complex Ω(X) of X. In consequence, the algebra generators of the real cohomology of LX can be represented by differential form...

In terms of generators and defining relations, a description is given of the Hochschild cohomology algebra for one of the series of local algebras of quaternion type. As a corollary, the Hochschild cohomology algebra is described for the group algebras of generalized quaternion groups over algebraically closed fields of characteristic 2. Introduction Let R be a finite-dimensional algebra over a...

in this paper, we consider a class of connected oriented (with respect to z/p) closed g-manifolds with a non-empty finite fixed point set, each of which is g-equivariantly formal, where g = z/p and p is an odd prime. using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a g-manifold in terms of algebra. this makes it p...

For a family of local algebras of semidihedral type over an algebraically closed field of characteristic 2, the Hochschild cohomology algebra is described in terms of generators and relations. The calculations are based on the construction of a bimodule resolution for the algebras in question. As a consequence, the Hochschild cohomology algebra is described for the group algebras of semidihedra...

In this appendix we give another algebraic proof of the Weyl character formula, using methods that have many other applications in Lie theory. We begin by setting up the machinery of Lie algebra cohomology (without assuming any previous background in homological algebra). We define the cohomology spaces for a Lie algebra representation in terms of a cochain complex and differential, with the co...