Tensor categories, quantized algebras, and the analytic Langlands correspondence

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, for example groups, Lie algebras, and 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. This project will study ordinary and tensor categories, some of which arise as representation categories and some of which don't, and connections between them. In particular, the PI will study non-integer 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 non-integer. This 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 project also involves the study of 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. Finally, the project will continue tthe study of the analytic Langlands correspondence, which was initiated by the PI with E. Frenkel and D. Kazhdan. This is a new subject that unifies several topics of current interest in algebra, number theory, geometry, and quantum physics. The project also provides research training opportunities for graduate students and the PI will supervise the work of high school students in MIT PRIMES.

In more detail, the PI plans to: 1) Continue to develop Lie theory in tensor categories in positive characteristic, in particular the Verlinde category Ver(p); study and classify simple and linearly reductive Lie algebras in this category, compute their cohomology and study representations; 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 a prime p; compute the cohomology of higher Verlinde categories Ver(p^n); 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; continue to develop the theory of actions of finite dimensional Hopf algebras on division algebras, and in particular, fields; and classify unipotent tensor categories. 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), or Sp(n)) to non-integer values of the parameter n; compute reducibility loci and obtain various character formulas and signature formulas in these representation theories, and answer various other representation theoretic questions; study similar questions in the recently introduced Delannoy and arboreal tensor categories. 3) Study signatures of representations of quantum groups and Hecke algebras for |q|=1 and of Cherednik algebras; work on a discrete analog of the monodromy theorem for the Casimir connection; work on the representation theory of deformed double current algebras, representations of cyclotomic Cherednik algebras, representations of Cherednik algebras in positive characteristic, direct and inverse image functors for Cherednik algebras, short star-products on quantizations, centers of quantum affine algebras when the level parameter is a root of unity. 4) Continue to work with E. Frenkel and D. Kazhdan on the analytic Langlands correspondence and explore applications of separation of variables.

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.

Award Number: 2502467
Principal Investigator: Pavel Etingof
Funds Obligated: $267,000
State: MA