Tensor Categories and Representation Theory

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Representation theory is a study of symmetries of space, such as our 3-dimensional space, or, more generally, a space with any number of dimensions (even infinite). In this theory, symmetries of the underlying space are encoded in an algebraic structure and the elements in the algebraic structure are represented by linear transformations, or, more explicitly, by matrices. Thus, a representation is basically a collection of matrices that satisfy a certain natural system of nonlinear equations. These equations are determined by the collection of symmetries that are being studied. Representations of a given structure themselves form a quite intricate and rich structure, which encodes relations (or mappings) between different representations. This higher-level structure is called the category of representations. For some types of structures (e.g., for groups, Lie algebras, and quantum groups), representations can be multiplied; in this case the corresponding categories are tensor categories because multiplication of representations is similar to multiplication of tensors. It turns out that the notion of a tensor category is very interesting in its own right, and that many tensor categories don't arise as categories of representations. This research concerns ordinary and tensor categories, some of which arise as representation categories and some of which don't, and to study connections between them. In particular, the project studies complex rank generalizations of representation categories proposed by P. Deligne. Roughly speaking, this is a generalization in which the number of elements of a set or rows of a matrix is allowed to be nonintegral. This becomes meaningful and useful when the invariants of interest turn out to be polynomials of the number of elements or rows, which is often the case. The project also involves the study of quantum groups which describe hidden symmetries of quantum systems, and yield tensor categories which lead to invariants allowing us to distinguish in-equivalent knots and links. This research project will study tensor categories; Hopf algebra actions on rings; quantum groups; representation theory in complex rank; Hecke algebras, Cherednik algebras, symplectic reflection algebras; noncommutative algebra; and Poisson homology. The PI's plan is as follows. (1) Develop a theory of actions of finite dimensional Hopf algebras on division algebras (in particular, fields) and apply it to proving non-existence statements for Hopf actions, develop a theory of extensions of tensor categories, classify unipotent categories, and classify fiber functors and module categories for the small quantum group. (2) Study a discrete analog of the monodromy theorem of Toledano Laredo for the Casimir connection, using dynamical Weyl groups; trace functions for quantum affine algebras; and signatures of representations of quantum groups for |q|=1. (3) Develop the ideas of P. Deligne, and extend representation theories of various classical structures (containing the symmetric group S(n) or classical Lie groups GL(n), O(n), Sp(2n)) to complex values of n. (4) Study the representation theory of double Yangians, of Cherednik algebras on curves, on elliptic algebras, and on signatures of representations of rational Cherednik algebras. (5) Supervise the work of undergraduate and high school students on the lower central series of associative algebras.