The Lie algebra of derivations of a block of a finite group

李群和李代数在物理学中是作为物理系统及其切线空间的对称群而产生的，它们可以被看作是无限小的对称运动。这些概念在许多数学领域一直是人们感兴趣的。例如，李代数是作为遵守莱布尼茨乘积规则的代数上的算子而产生的，称为导子。代数上的导子可以解释为代数的第一个Hochschild上同调的代表。这个空间上的李代数结构推广到Hochschild上同调上的分次李代数结构--这可以追溯到Gersten haber在代数形变理论的背景下的开创性工作。这一技术在物理学中的应用往往是在特征为零的域上，但其基本概念在素特征域上具有类似的性质，并使该技术可用于局部环和域上有限群代数模表示理论的研究。事实上，素数特征域上的有限维李代数比复数域上的有限维李代数要多得多。有限群的模表示理论的一个特殊特征是它由大量的猜想驱动，其中一些猜想预测了有限群代数的各种直接因子之间的显著结构联系，另一些则简单地预测了神秘的数值巧合。通常情况下，Hochschild上同调被证明是有用的，用于这些猜想的重构和变形。在有限群代数及其直接因子的背景下研究Hochschild上同调的(分次的和受限的)李代数结构将有助于理解这些猜想的某些部分。本文正是以这些期望为出发点，把重点放在了Hochschild上同调的更高结构方面及其对有限群代数及其块的不变量的影响。我们开始用九个猜想的序列来描述这个程序，范围从基本问题--例如块的第一个Hochschild上同调不为零--通过某些有限群类的显式计算，到目前关于有限群及其块的数值和结构方面的难以捉摸的猜想。

Lie groups and Lie algebras arise in Physics as symmetry groups of physical systems and their tangent spaces, which may be regarded asinfinitesimal symmetry motions.These notions have long been of interest in many areas of Mathematics.Lie algebras arise, for instance, as operators on algebras respectingLeibniz' product rule, called derivations. The derivations on an algebra can be interpreted as representatives ofthe first Hochschild cohomology of an algebra. The Lie algebra structureon this space extends to a graded Lie algebra structure on Hochschildcohomology - this goes back to pioneering work of Gerstenhaber, in thecontext of the deformation theory of algebras.The use of this technology in Physics tends to be over fields ofcharacteristic zero, but the underlying concepts have analogues over fields of prime characteristic, and makes this technology available for investigations in the modular representation theory of finite group algebras over local rings and fields. In fact, there are `many more' finite-dimensional Lie algebras over fields of prime characteristic thanover the complex numbers.A particular feature of modular representation theory of finite groupsis that it is driven by a great number of conjectures, some of which predict remarkable structural connection between various direct factors of finite group algebras, and other simply predicting mysterious numerical coincidences.Hochschild cohomology in general has turned out to be useful forreformulations and variations of those conjectures. Expectations arehigh that investigating the (graded and restricted) Lie algebra structure of Hochschild cohomology in the context of finite group algebras and their direct factors should contribute to an understanding of some parts of those conjectures. The present proposal takes precisely these expectations as a startingpoint, putting the focus on higher structural aspects ofHochschild cohomology and their impact on invariants of finite group algebras and their blocks. We set out describing this programme in a sequence of nine conjectures, ranging from basic questions - such as the non-vanishing of the first Hochschild cohomology of blocks - viaexplicit calculations in certain classes of finite groups to currently elusive conjectures on numerical and structural aspects offinite groups and their blocks.