One of our main results is a classification all the possible quivers of selfinjective radical cube zero finite dimensional algebras over an algebraically closed field having finite complexity. In the paper [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite dimensional algebra to have such a theory of support var...

We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when M is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer & Greenlee...

We exploit the Fedosov-Weinstein-Xu (FWX) resolution proposed in q-alg/9709043 to establish an isomorphism between the ring of Hochschild cohomology of the quantum algebra of functions on a symplectic manifold M and the ring H(M,C((~))) of De Rham cohomology of M with the coefficient field C((~)) without making use of any version of formality theorem. We also show that the Gerstenhaber bracket ...

We set up a fibred categorical theory of obstruction and classification morphisms that specialises to the one monoidal functors between groups also Schreier-Mac Lane group extensions. Further applications are provided crossed extensions bimodule butterflies, with in particular non-abelian unital associative algebras terms Hochschild cohomology.

We compute the Hochschild cohomology and homology of the algebra Λ = k〈x, y〉/(x2, xy + qyx, y2) with coefficients in 1Λψ for every degree preserving k-algebra automorphism ψ : Λ → Λ. As a result we obtain several interesting examples of the homological behavior of Λ as a bimodule.

Given a conditionally completely positive map L on a unital ∗-algebra A, we find an interesting connection between the second Hochschild cohomology of A with coefficients in the bimodule EL = Ba(A⊕M) of adjointable maps, where M is the GNS bimodule of L, and the possibility of constructing a quantum random walk (in the sense of [2, 11, 13, 16]) corresponding to L.

Let X be a simply connected space and LX the space of free loops on X. We determine the mod p cohomology algebra of LX when the mod p cohomology of X is generated by one element or is an exterior algebra on two generators. We also provide lower bounds on the dimensions of the Hodge decomposition factors of the rational cohomology of LX when the rational cohomology of X is a graded complete inte...