Tensor Categories and Representations of Quantized Algebras

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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 (even infinite number) of dimensions. In this theory, symmetries are represented by linear transformations of this space, or, more explicitly, by matrices. Thus, a representation of a given symmetry structure is basically a collection of matrices which satisfy a certain natural system of nonlinear equations. The equations are determined by the exact type of symmetry structure we are representing - a group, a Lie algebra, or an associative algebra. 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 type of structures (e.g. for groups, Lie algebras, quantum groups), representations can be multiplied; in this case the corresponding categories are tensor categories (as 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. The PI will investigate ordinary and tensor categories, some of which arise as representation categories and some of which don't, as well as the connections between them. In particular, complex rank generalizations of representation categories proposed by P. Deligne will be investigated. Roughly speaking, this is a generalization in which the number of elements of a set or rows of a matrix is allowed to be non-integer. This seemingly nonsensical setting becomes meaningful and useful when the invariants one is interested in turn out to be polynomials of the number of elements or rows, which is often true. The PI will also investigate quantizations of singular symplectic varieties, for instance symplectic resolutions. These are non-commutative algebras that appear in certain kinds of quantum field theories of recent interest as algebras of quantum observables. This project provides research training opportunities for graduate students.This project involves research on: tensor categories; quantum groups; representation theory in complex rank; cherednik algebras; short star-products on quantizations; analytic approach to Geometric Langlands program. The plan of PI's work is as follows. 1. Develop a theory of Frobenius functors for symmetric tensor categories in characteristic p and Frobenius exact categories; classify exact factorizations of fusion categories, in particular twisted Deligne products; classify fiber functors and module categories over the representation category of the small quantum group; compute the semisimplification of the category of tilting modules for a reductive group in small characteristic, and use it to compute the dimensions of tilting modules modulo p; prove quasi-motivicity of representations of braid groups arising from braided fusion categories; construct new symmetric tensor categories in characteristic p2 similar to the Etingof-Benson categories in characteristic 2; compute cohomology of these categories; develop Lie theory in the Verlinde category; develop a theory of symplectic reflection fusion categories; continue to develop the theory of actions of finite dimensional Hopf algebras on division algebras (in particular, fields); classify unipotent tensor categories. Work on a discrete analog of the monodromy theorem of Toledano Laredo for the Casimir connection, using dynamical Weyl groups, Study signatures of representations of quantum groups for |q|=1. 2. Continue to 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 the rank parameter n. These structures will include degenerate affine Hecke algebras, rational and trigonometric Cherednik algebras, symplectic reflection algebras, real reductive Lie groups (i.e., symmetric pairs), Lie superalgebras, affine Lie algebras, (parabolic) category O for reductive Lie algebras, Yangians, and other structures. Compute reducibility loci and obtain various character formulas and signature formulas in these representation theories, and answer various other representation theoretic questions which are known to be interesting in the classical setting. 3. Work on the representation theory of double Yangians, the theory of elliptic algebras, representations of cyclotomic Cherednik algebras, signatures of representations of Cherednik algebras, representations of Cherednik algebras in positive characteristic, direct and inverse image functors for Cherednik algebras. 4. Continue to develop the theory of short star-products on filtered quantizations. 5. Continue to work with E. Frenkel and D. Kazhdan on an analytic approach to the geometric Langlands correspondence.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.