## Invariance of the p-power map under stable equivalences of Morita type

### 报告题目(Title)：**Invariance of the p-power map under stable equivalences of Morita type**

报告人(Speaker)：*Lleonard Rubioy Degrassi （西班牙Murcia大学）*

地点(Place)：*教八楼209教室；后主楼1223报告厅；教八楼209教室*

时间(Time)：*2019年9月26日16:00-17:10；9月28日9:00-10:10；10月10日16:00-17:10*

邀请人(Inviter)：*刘玉明*

### 报告摘要

Let A be a finite-dimensional associative unital algebra over a field. The Hochschild cohomology of A is a Gerstenhaber algebra, that is, a graded-commutative algebra, under the cup product, with a Lie bracket of degree -1, also called Gerstenhaber bracket, satisfying the Poisson identity. In particular, its first degree component, denoted by HH^{1}(A), is a Lie algebra. In addition, in positive characteristic HH^{1}(A) is a restricted Lie algebra.

In this series of three lectures, we will examine in detail how the restricted Lie algebra structure of HH^{1}(A) is related with the representation theory of A. We will also see applications in modular representation theory of finite groups.

First talk: Invariance of the p-power map under stable equivalences of Morita type

Symmetric algebras are usually classified by Morita or derived equivalences or by stable equivalences of Morita type. Stable equivalences of Morita type are the most general and frequent in modular representation theory but also the least understood. The main reason is that we do not know if many of the invariants for Morita and derived equivalences are still invariants for stable equivalences of Morita type.

In this context, the first degree component of Hochschild cohomology plays a crucial rule: its Lie algebra structure is an invariant under stable equivalences of Morita type. A natural questions emerges: is in positive characteristic the restricted Lie algebra structure of HH^{1}(A) also preserved under stable equivalences of Morita type?

In this talk we will show that this holds for a subclass of HH^{1}(A) called integrable derivation: these are derivations that are coming from automorphisms on A[[t]] inducing the identity on A[[t]]/t^rA[[t]]. Using transfer maps as our main tool, we will show the following: the p-power map, restricted to the classes of integrable derivations, commutes with stable equivalences of Morita type between finite-dimensional selfinjective algebras.

Second talk: On the simplicity of the first Hochschild cohomology in modular representation theory

Every finite-dimensional algebra is decomposed as a direct sum of two-sided ideals called blocks. In modular representation theory to each block (of group algebra kG of a finite group G) we assign a p-subgroup of G called defect group which measures how far the block is from being semisimple. The block decomposition allows us also to study blocks depending on the number of simple kG-modules. However, even in the case of blocks with one simple module, the situation is poorly understood.

In the second talk we will show how the Lie structure of HH^{1}(A) helps to understand blocks with one simple module. More precisely, let B be a block with one simple module. Then HH^{1}(A) is a simple Lie algebra if and only if B is nilpotent (Morita equivalent to the group algebra of its defect group) with an elementary abelian defect group P of order at least 3. This is joint work with Markus Linckelmann.

Third talk: On the solvability of first Hochschild cohomology

Let A be a finite-dimensional algebra over an algebraically closed field. A famous theorem of Gabriel states that there is a bijection between the isomorphism classes of simple A-modules and the number of vertices of the quiver, and between the number of arrows of the quiver and the dimensions of the extensions between simple A-modules.

In the last talk I will show how the Lie structure of HH^{1}(A) is strongly related with the Ext-quiver of A. More precisely, if we assume that the Ext-quiver of A is a simple directed graph, then HH^{1}(A) is a solvable Lie algebra. I will also show that in the case of having at most two parallel arrows in the Ext-quiver of A but no loops, then the only simple Lie algebra that could arise is sl_2(k). For quivers containing loops, I will determine sufficient conditions for the solvability of HH^{1}(A). Finally, I will apply these criteria to show the solvability of the first Hochschild cohomology of blocks with cyclic defect, all tame blocks of finite groups and some wild algebras. This is part of two recent joint works with Markus Linckelmann, and with Andrea Solotar and Sibylle Schroll.

In this series of three lectures, we will examine in detail how the restricted Lie algebra structure of HH^{1}(A) is related with the representation theory of A. We will also see applications in modular representation theory of finite groups.

First talk: Invariance of the p-power map under stable equivalences of Morita type

Symmetric algebras are usually classified by Morita or derived equivalences or by stable equivalences of Morita type. Stable equivalences of Morita type are the most general and frequent in modular representation theory but also the least understood. The main reason is that we do not know if many of the invariants for Morita and derived equivalences are still invariants for stable equivalences of Morita type.

In this context, the first degree component of Hochschild cohomology plays a crucial rule: its Lie algebra structure is an invariant under stable equivalences of Morita type. A natural questions emerges: is in positive characteristic the restricted Lie algebra structure of HH^{1}(A) also preserved under stable equivalences of Morita type?

In this talk we will show that this holds for a subclass of HH^{1}(A) called integrable derivation: these are derivations that are coming from automorphisms on A[[t]] inducing the identity on A[[t]]/t^rA[[t]]. Using transfer maps as our main tool, we will show the following: the p-power map, restricted to the classes of integrable derivations, commutes with stable equivalences of Morita type between finite-dimensional selfinjective algebras.

Second talk: On the simplicity of the first Hochschild cohomology in modular representation theory

Every finite-dimensional algebra is decomposed as a direct sum of two-sided ideals called blocks. In modular representation theory to each block (of group algebra kG of a finite group G) we assign a p-subgroup of G called defect group which measures how far the block is from being semisimple. The block decomposition allows us also to study blocks depending on the number of simple kG-modules. However, even in the case of blocks with one simple module, the situation is poorly understood.

In the second talk we will show how the Lie structure of HH^{1}(A) helps to understand blocks with one simple module. More precisely, let B be a block with one simple module. Then HH^{1}(A) is a simple Lie algebra if and only if B is nilpotent (Morita equivalent to the group algebra of its defect group) with an elementary abelian defect group P of order at least 3. This is joint work with Markus Linckelmann.

Third talk: On the solvability of first Hochschild cohomology

Let A be a finite-dimensional algebra over an algebraically closed field. A famous theorem of Gabriel states that there is a bijection between the isomorphism classes of simple A-modules and the number of vertices of the quiver, and between the number of arrows of the quiver and the dimensions of the extensions between simple A-modules.

In the last talk I will show how the Lie structure of HH^{1}(A) is strongly related with the Ext-quiver of A. More precisely, if we assume that the Ext-quiver of A is a simple directed graph, then HH^{1}(A) is a solvable Lie algebra. I will also show that in the case of having at most two parallel arrows in the Ext-quiver of A but no loops, then the only simple Lie algebra that could arise is sl_2(k). For quivers containing loops, I will determine sufficient conditions for the solvability of HH^{1}(A). Finally, I will apply these criteria to show the solvability of the first Hochschild cohomology of blocks with cyclic defect, all tame blocks of finite groups and some wild algebras. This is part of two recent joint works with Markus Linckelmann, and with Andrea Solotar and Sibylle Schroll.