Guodong Zhou

The Hall algebra of a spherical object

We determine the Hall algebra, in the sense of B. To¨en, of the algebraic triangulated category generated by a spherical object.

Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander–Reiten conjecture

Derived equivalences and stable equivalences of Morita type, and new (candidate) invariants thereof, between symmetric algebras will be investigated, using transfer maps as a tool. Close relationships will be established between the new invariants and the validity of the Auslander–Reiten conjecture, which states the invariance of the number of non-projective simple modules under stable equivalence. More precisely, the validity of this conjecture for a given pair of algebras, which are stably equivalent of Morita type, will be characterized in terms of data refining Hochschild homology (via Külshammer ideals) being invariant and also in terms of cyclic homology being invariant. Thus, validity of the Auslander–Reiten conjecture implies a whole set of ring theoretic and cohomological data to be invariant under stable equivalence of Morita type, and hence also under derived equivalence. We shall also prove that the Batalin-Vilkovisky algebra structure of Hochschild cohomology for symmetric algebras is preserved by derived equivalence. The main tools to be developed and used are transfer maps and their properties, in particular a crucial compatibility condition between transfer maps in Hochschild homology and Hochschild cohomology via the duality between them.

The Batalin–Vilkovisky structure over the Hochschild cohomology ring of a group algebra

We realize explicitly the well-known additive decomposition of the Hochschild cohomology ring of a group algebra in the elements level. As a result, we describe the cup product, the Batalin-Vilkovisky operator and the Lie bracket in the Hochschild cohomology ring of a group algebra.