Integrable derivations and Hochschild cohomology of block algebras of finite groups

真实的值函数求导运算的乘积法则在代数学中有着广泛的推广和应用。它导致的概念导子代数-theseeare线性自同态代数satifying产品规则。它们表示代数的第一Hochschild上同调类。代数的第一Hochschild上同调是一个李代数，更确切地说，是一个限制李代数，如果底层基环是一个正特征域。有限群的模表示理论试图理解有限群的结构和相关的群代数之间的联系。许多驱动这一领域的假设是-迄今为止神秘的-数字巧合有关的不变量的有限群代数的不变量的基础群体。在前面的段落中暗示的复杂的上同调技术预计会产生一些关于这些同伦的见解，本建议将重点放在某些可积导子群在Morita下的一些精确的和未探索的不变性质上，导出，或有限群代数的不可分解代数因子之间的稳定等价，它们的特征标理论，它们的自同构群，和有限群的局部结构。

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.