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空间的对称群出现的，它们可以被看作是无穷小的对称运动。这些概念长期以来在数学的许多领域都引起了人们的兴趣。例如，李代数作为代数上符合莱布尼茨乘积法则的算子而出现，这个法则被称为派生。代数上的导数可以解释为代数的第一个Hochschild上同调的表示。这个空间上的李代数结构扩展到hochschild上同调上的阶李代数结构——这可以追溯到Gerstenhaber在代数变形理论的背景下的开创性工作。该技术在物理中的应用往往是在特征为零的场上，但其基本概念在素数特征的场上有类似的应用，并使该技术可用于局部环和场上有限群代数的模表示理论的研究。事实上，在素数特征域上的有限维李代数比在复数域上的“多得多”。有限群的模表示理论的一个特点是它是由大量的猜想驱动的，其中一些猜想预测了有限群代数的各种直接因子之间显著的结构联系，而另一些则简单地预测了神秘的数值巧合。霍克希尔德上同调通常被证明是这些猜想的有用的重新表述和变化。在有限群代数及其直接因素的背景下，研究Hochschild上同调的（分级和限制）李代数结构应该有助于理解这些猜想的某些部分。目前的建议正是以这些期望为出发点，把重点放在hochschild上同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，从基本的问题——比如块的第一个Hochschild上同调的不消失——通过某些类有限群的显式计算，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.