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

李群和李代数起源于物理学中，作为物理系统及其切空间的对称群，它们可以被看作是无穷小的对称运动。这些概念在数学的许多领域都很有意义。例如，李代数起源于代数上的算子，它们遵循莱布尼茨的乘积规则，称为导子。代数上的导子可以解释为代数的第一Hochschild上同调的代表。这个空间上的李代数结构扩展到Hochschild上同调的分次李代数结构-这可以追溯到Gerstenhaber在代数变形理论的背景下的开创性工作。在物理学中使用这种技术往往是在特征为零的领域上，但基本概念在素特征的领域上有类似之处，并使这种技术可用于局部环和域上有限群代数的模表示理论的研究。事实上，素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点是它是由大量的定理驱动的，其中一些定理预言了有限群代数的各种直接因子之间的显著结构联系，和其他简单地预测神秘的数字巧合。Hochschild上同调一般已被证明是有用的forreformations和变化的那些结构。期望很高，调查（分次和限制）李代数结构的Hochschild上同调的有限群代数及其直接因素的背景下，应有助于理解这些结构的某些部分。本提案正是以这些期望为出发点，把重点放在更高的结构方面ofHochschild上同调及其对有限群代数及其块的不变量的影响。我们开始描述这一计划在一个序列的九个aptures，范围从基本的问题-如非零的第一Hochschild上同调块-通过明确的计算在某些类的有限群目前难以捉摸的aptures的数值和结构方面ofinite集团和他们的块。

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.