Integrable derivations and Hochschild cohomology of block algebras of finite groups

我们所熟悉的实值函数求导运算的乘积法则在代数中有大量的推广和应用。它引出了代数的导数的概念——代数的线性自同态满足乘积法则。它们表示代数的第一Hochschild上同的类。一个代数的第一个Hochschild上同调被证明是一个李代数，更准确地说，如果下基环是一个正特征域，则是一个受限李代数。（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代数变形的开创性工作。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。推动这一领域的许多猜想——迄今为止是神秘的——是有限群代数的不变量与底层群的不变量之间的数值巧合。上一段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，而本建议将重点放在有限群代数的不可分解代数因子之间的Morita，导出或稳定等价下的某些可积导数群的一些精确和未探索的不变性，它们的特征理论，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.