This project proposes research on tensor categories; quantum groups; representation theory in complex rank; Hecke algebras, Cherednik algebras, symplectic reflection algebras; noncommutative algebra; Poisson homology. The PI's work plan is as follows. 1) Continue developing the general theory of finite tensor categories, in particular, of fusion categories. 2) Prove a discrete analog of the monodromy theorem of Toledano Laredo for the Casimir connection, using dynamical Weyl groups, prove the Felder-Varchenko conjectures on trace functions, and study the trigonometric, the discrete, and the discrete trigonometric analogs of the quantum shift-of-the-argument algebra. 3) Study finite dimensional representations of rational Cherednik algebras and symplectic reflection algebras, representations of continuous Hecke algebras, unitary representations of Cherednik algebras. Study various constructions which link representation theory of these algebras with Lie theory and the representation theory of quantum groups. 4) Develop the ideas of P. Deligne, and extend representation theories of various classical structures (containing the symmetric group or classical Lie groups) to complex values of the rank parameter n (these structures include degenerate affine Hecke algebras, rational and trigonometric Cherednik algebras, symplectic reflection algebras, real reductive Lie groups, Lie superalgebras, affine Lie algebras, parabolic category O for reductive Lie algebras, Yangians, and other structures). 5) Work on quantizations of multiplicative quiver varieties, and on the structure of the lower central series of associative algebras. 6) Continue to study the structure of the zeroth Poisson homology of Poisson varieties.
Representation theory is a study of symmetry in a vector space. In this theory, symmetries are represented by linear transformations of this space (by matrices). Thus, a representation of a given symmetry structure is basically a collection of matrices which satisfy a certain natural system of nonlinear relations. The relations are determined by the exact type of structure to be represented such as a group, a Lie algebra, or an associative algebra. A higher-level structure is called the category of representations. For some type of structures (e.g. for groups, Lie algebras, quantum groups), representations can be multiplied to form tensor categories. The present project proposes to study many ordinary and tensor categories, some of which arise as representation categories and some of which don't, and to study connections between them. The PI proposes to study complex rank generalizations of representation categories proposed by Deligne. Roughly speaking, this is a generalization in which the number of rows of a matrix is allowed to be non-integer. This seemingly nonsensical setting becomes meaningful and useful in a situation when the interesting invariants are polynomials of the number of rows.
