Hochschild Cohomology of Algebras: Structure and Applications

Speaker: Petter Andreas Bergh (NTNU) Title: Hochschild (co)homology of quantum complete intersections Abstract: This is joint work with Karin Erdmann. We construct a minimal projective bimodule resolution for finite dimensional quantum complete intersections of codimension 2. Then we use this resolution to compute the Hochschild homology and cohomology for such an algebra. In particular, we show that when the commutator element is not a root of unity, then the cohomology vanishes in high degrees, while the homology is always nonzero. Thus these algebras provide further counterexamples to “Happel’s question”, a question for which the first counterexample was given by Buchweitz, Green, Madsen and Solberg. On the other hand, the homology of the quantum complete intersections behave in accordance with Han’s conjecture, i.e. the homology version of Happel’s question. Speaker: Emily Burgunder (Université de Montpellier II) Title: Leibniz homology and Kontsevich’s graph complexes Abstract: The homology of the Lie algebra of matrices gl(A) over an associative algebra A can be computed thanks to the cyclic homology of A: H(gl(A)) = S(HC(A)) This theorem is known as the Loday-Quillen-Tsygan theorem. Another well-known theorem in homology is due to Kontsevich which says that the homology of the symplectic Lie algebra K[p1, ...pn, q1, ...qn] can be explicited thanks to the homology of a certain “graph complex” G: H(K[p1, ..., pn, q1, ...qn]) = S(H(G)) These two theorems are, in fact, examples of a more general theorem in the operadic setting, that we will present. If we replace the Lie homology by the Leibniz homology, then the cyclic homology has to be replaced by the Hochschild homology (Cuvier-Loday theorem). We show that, in Kontsevich case, there exists a “nonsymmetric graph complex” which computes the Leibniz homology of the symplectic Lie algebra. Speaker: Claude Cibils (Université de Montpellier II) Title: The Intrinsic Fundamental Group of a Linear Category Abstract: Joint work with Maria Julia Redondo and Andrea Solotar. The main purpose is to provide a positive answer to the question of the existence of an intrinsic and canonical fundamental group associated to a linear category. The fundamental group we introduce takes into account the linear structure of the category, it differs from the fundamental group of the underlying category obtained as the classifying space of its nerve (see for instance G. Segal [1968] or D. Quillen [1973]). We provide an intrinsic definition of the fundamental group as the automorphism group of the fibre functor on Galois coverings. We prove that this group is isomorphic to the inverse limit of the Galois groups associated to Galois coverings. Moreover, the graduation deduced from a Galois covering enables