Noncommutative Representation Theory

对称性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(或表示)来理解和量化对称性。例如，许多动物的两侧对称性是由反射操作编码的，而一个人需要旋转和反射来描述晶体更精细的对称性。在某些环境中，如量子物理，尽管具有一定的对称性，但其固有的非对易性有时可以通过推广对称群的Hopf代数的作用来看待。本课题主要研究Hopf代数的表示及其相关的代数结构问题。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，关于它们的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，并为许多与PI合作的研究生、博士后和其他年轻数学家提供了很好的培训基础。Hopf代数的表示范畴是张量范畴，由它们的余积产生的丰富结构。与群相比，Hopf代数的行为非常不同，令人惊讶。与群的张量范畴相比，表示的张量范畴可以是非对易的，这导致了一些奇怪的行为。这个项目研究支持变化理论和这些非交换表示范畴的三角范畴结构，目的是更完整地发展它，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，该项目还需要提高对(有限维)Hopf代数的同调理解，因为上同调被猜想是有限生成的。在与合作者的合作中，PI的目的是证明一些重要的Hopf代数类的猜想，即包括作为特例的小量子群的对角型定点Hopf代数。与更好地理解的小量子群相比，在这类更大的Hopf代数中，许多可能具有非交换表示范畴，人们必须在同调工作中更加小心，因为这会导致细微的差异。一些相关的同调项目涉及一个Hopf代数在一个代数上的作用，这可以通过构造一个由这两个代数构成的更大的代数来研究。PI将研究这样的sash乘积代数和它们的Hochschild上同调，特别是Gertenhaber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.