Tamarkin-Tsygan calculus for gentle algebras

Abstract: The whole structure given by the Hochschild cohomology and homology of an associative algebra A together with the cup and cap products, the Gerstenhaber bracket and the Connes differential is called the Tamarkin-Tsygan calculus. It is invariant under derived equivalence and if we can compute all these invariants provides a lot of information. The calculation of the whole Tamarkin-Tsygan calculus is very difficult and generally not even possible for particular algebras. However, there exist some calculations for individual algebras. The problem is, in general, that the minimal projective bimodule resolutions are difficult to find and even if one is able to compute such a resolution, it might be so complicated that the computation of the Tamarkin-Tsygan calculus is not within reach. For monomial algebras the minimal projective bimodule resolution is known and in the case of quadratic monomial algebras it is simple enough, to embark on the extensive calculations of the Tamarkin Tsygan calculus. Yet even for quadratic monomial algebras, the combinatorial level of the calculations is such that it is too complicated to calculate the whole calculus. On the other hand for gentle algebras, the additional constraints on their structure are such that the calculations become possible. We will focus on the concrete aspects of these calculations.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar