Maximal tori in HH^1 and the fundamental group(s)
报告题目(Title)：Maximal tori in HH^1 and the fundamental group(s)
报告人(Speaker)：Dr. Lleonard Rubio Y Degrassi (University of Verona)
地点(Place)：Zoom ID: 844 521 31211 Passcode: 202112
时间(Time)：December 23, 2021, 16:00-17:00
邀请人(Inviter)：胡维
报告摘要
Hochschild cohomology is a fascinating invariant of an associative algebra which possesses a rich structure. In particular, the first Hochschild cohomology group $HH^1(A)$ of an algebra $A$ is a Lie algebra, which is a derived invariant and, among self-injective algebras, an invariant under stable equivalences of Morita type. This establishes a bridge between finite dimensional algebras and Lie algebras, however, aside from few exceptions, fine Lie theoretic properties of $HH^1(A)$ are not often used.
In this talk, I will show some results in this direction. More precisely, I will explain how maximal tori of $HH^1(A)$, together with fundamental groups associated to presentations of $A$, can be used to deduce information about the shape of the Gabriel quiver of $A$. In particular, I will show that every maximal torus in $HH^1(A)$ arises as the dual of some fundamental group of $A$. By combining this, with known invariance results for Hochschild cohomology, I will deduce that (in rough terms) the largest rank of a fundamental group of $A$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Time permitting, I will also provide various applications to semimonomial and simply connected algebras.
This is joint work with Benjamin Briggs.
