Noncommutative Representation Theory

对称在自然界和数学中无处不在。数学家通过对称群及其对物体的作用（或表现）来理解和量化对称。例如，许多动物的双边对称性是通过反射运算来编码的，而人们需要旋转和反射来描述晶体更精细的对称性。在某些情况下，如量子物理中固有的非交换性，尽管具有一定的对称性，但有时可以通过推广对称群的Hopf代数的作用来观察。本课题研究Hopf代数的表示和相关代数结构的问题。Hopf代数出现在许多数学领域，如组合学、数论、拓扑学和数学物理，其表示的结果引起了广泛的兴趣。该研究项目涉及许多合作者，并为PI工作的许多研究生，博士后和其他年轻数学家提供了良好的训练基地。Hopf代数表示的范畴是张量范畴，一个由它们的余积产生的丰富结构。与群相比，Hopf代数可以表现得非常不同，令人惊讶。与群的张量范畴相比，表示的张量范畴可以是非交换的，这导致了一些奇怪的行为。本项目研究了这些非交换范畴的表示的支持变理论和三角化范畴结构，其目标是更全面地发展它，就像在有限群和有限群方案的（交换）情况下所做的那样。为了取得这样的进展，该项目还需要推进（有限维）Hopf代数的同调理解，其上同调被推测为有限生成。在与合作者的合作中，PI旨在证明一些重要的Hopf代数类的猜想，即对角型的尖Hopf代数，其中包括作为特例的小量子群。与更好理解的小量子群相比，这类更大的Hopf代数中的许多可能具有非交换的表示范畴，并且在同调工作中必须更加小心，因为这会导致微妙的差异。一些相关的同调工程涉及一个Hopf代数对另一个代数的作用，这可以通过构造一个由这两个代数组成的更大的代数来研究。PI将研究这些粉碎积代数及其Hochschild上同调，特别是Gerstenhaber代数结构，以及相关的变形。

Symmetry is ubiquitous in nature and in mathematics. Mathematicians understand and quantify symmetry via symmetry groups and their actions (or representations) on objects. For example, the bilateral symmetry of many animals is encoded by a reflection operation, while one needs both rotations and reflections to describe the more elaborate symmetry of a crystal. The noncommutativity inherent in some settings such as quantum physics, that nevertheless possess some symmetry, is sometimes viewed via actions of the Hopf algebras that generalize symmetry groups. This project investigates questions about representations of Hopf algebras and related algebraic structures. Hopf algebras arise in many fields of mathematics, such as combinatorics, number theory, topology, and mathematical physics, and results on their representations are of wide interest. The research project involves many collaborators and serves as a good training ground for the many graduate students, postdocs, and other young mathematicians with whom the PI works.Categories of representations of Hopf algebras are tensor categories, a rich structure arising from their coproducts. In comparison to groups, Hopf algebras can behave quite differently and surprisingly. Tensor categories of representations can be noncommutative, in contrast to those of groups, leading to some curious behavior. This project investigates support variety theory and the triangulated category structure of these noncommutative categories of representations, with the goal of developing it much more completely, as has been done in the (commutative) case of finite groups and finite group schemes. In order to make such progress, the project will also need to advance homological understanding of (finite dimensional) Hopf algebras, whose cohomology is conjectured to be finitely generated. In work with collaborators, the PI aims to prove the conjecture for some important classes of Hopf algebras, namely the pointed Hopf algebras of diagonal type that include as a special case the small quantum groups. In contrast to the better-understood small quantum groups, many in this larger class of Hopf algebras may have noncommutative categories of representations, and one must take more care in homological work as this leads to subtle differences. Some related homological projects involve an action of a Hopf algebra on an algebra, which may be studied by constructing a larger algebra built out of the two. The PI will study such smash product algebras and their Hochschild cohomology, in particular the Gerstenhaber algebra structure, and related deformations.