Integrable derivations and Hochschild cohomology of block algebras of finite groups

我们所熟知的实值函数导数运算的乘积规则在代数中有大量的推广和应用。它引出了代数导子的概念--这些导子是满足乘积规则的代数的线性自同态。它们表示代数的第一个Hochschild上同调的类。代数的第一个Hochschild上同调是李代数，更确切地说，如果下面的基环是正特征域，则是限制李代数。(受限的)李代数结构在Hochschild上同调中扩展到所有正度-这可以追溯到Gerstenhaberon变形代数的开创性工作。有限群的模表示理论试图理解有限群的结构和相关群代数之间的联系。到目前为止，推动这一领域的许多猜想都是将有限群代数的不变量与基础群的不变量联系起来的数值巧合。上一段中所暗示的复杂的上同调技术有望对这些巧合产生一些见解，而本建议的重点是某些可积导子群在Morita、派生的或稳定的等价下的一些精确的和未开发的不变性，它们的特征标理论，它们的自同构群，以及有限群的局部结构。

The product rule of the all familiar operation of taking derivatives of real valued functions has a plethora of generalisations and applications in algebra. It leads to the notion of derivations of algebras - theseare linear endomorphisms of an algebra satifying the product rule. They represent the classes ofthe first Hochschild cohomology of an algebra. The first Hochschild cohomology of an algebraturns out to be a Lie algebra, and more precisely, a restricted Lie algebra if the underlyingbase ring is a field of positive characteristic. The (restricted) Lie algebra structure extends toall positive degrees in Hochschild cohomology - this goes back to pioneering work of Gerstenhaberon defornations of algebras.Modular representation theory of finite groups seeks to understand the connections betweenthe structure of finite groups and the associated group algebras. Many of the conjectures that drivethis area are - to date mysterious - numerical coincidences relating invariants of finitegroup algebras to invariants of the underlying groups. The sophisticated cohomological technology hinted at in the previous paragraph is expected to yield some insight regarding thesecoincidences, and the present proposal puts the focus on some precise and unexploredinvariance properties of certain groups of integrable derivations under Morita, derived, or stable equivalences between indecomposable algebra factors of finite group algebras, their character theory, their automorphism groups, and the local structure of finite groups.