Sam Qunell

Representation theory, affine quantum groups, KLR (quiver Hecke) algebras, categorification

My research uses new abstract mathematical techniques to study certain physical concepts in statistical mechanics. Many of the formulas that are important in statistical mechanics are actually "shadows" of a much deeper mathematical structure that isn't as obvious from the physical point of view. My goal is to make this connection as explicit as possible.

In statistical mechanics, there are many vector spaces that are important. These vector spaces and the linear operators on them provide a concise way to describe the possible states of a very large physical system and its total energy. Studying these vector spaces and the operators on them directly can be challenging. One modern approach for these kind of problems is "categorification". This means that we produce an abstract mathematical object, called a category, for which our vector space is like a "shadow". The category is a richer and more complex object than the vector space, and anything we can prove about the category will also "decategorify" to a proof about the vector space. My goal is to push these categorical techniques further to provide some new answers to some old physical questions.

I am a representation theorist who uses category theory to describe the representations of affine quantum groups. For starters, take your favorite complex semisimple Lie algebra, like sl2. This is not an associative algebra, but it is contained in a universal enveloping algebra, which is associative. We take a one parameter deformation of the defining formulas for this enveloping algebra to obtain our quantum groups. The quantum groups that I am most interested in are those associated to affine Lie algebras. The representations of affine quantum groups arise in many places. For me, the most important of these is in statistical mechanics. Solvable lattice models in statistical physics can be obtained from an "R-matrix", which is a matrix satisfying the (quantum) Yang-Baxter equation. Drinfeld proved that such R-matrices can be obtained from finite-dimensional representations of affine quantum groups. So, an important mathematical question is to study the category of finite-dimensional representations of affine quantum groups and to see what kinds of R-matrices can be produced here, or from any of several generalizations. My research focuses on using categorifications of quantum groups to study their representations in more detail. Khovanov-Lauda, and independently Rouquier, produced monoidal categories whose Grothendieck groups are isomorphic to (integral forms of) quantum groups. Many results from the structure and representation theory of quantum groups have an upgrade to the categorical level. My research looks at studying certain new infinite-dimensional representations of affine quantum groups from the categorical point of view, as well as producing new categorifications of related algebras. The categorical point of view gives new tools for studying these objects and often yields elegant answers to old questions.

My research looks at finding new categorifications of certain quantum algebras and at using categorical (KLR) techniques to prove things about these algebras. My most recent project (top publication in list below) found new constructions of prefundamental representations and proofs of their character formulas. These prefundamental representations are certain distinguished infinite-dimensional representations of the quantum groups associated to affine Borel algebras. The prefundamental representations were originally defined in terms of Drinfeld's loop presentation using limit formulas, so it is quite surprising that (at least in type A_n) much more explicit formulas can be given in terms of the Kac-Moody generators. The character formula for these prefundamental representations is, surprisingly, in the same denominator-product form as that of the Verma modules. Although combinatorial proofs of these formulas are known, I provide some new proofs by showing directly that the prefundamental representations are quotients of the positive part quantum groups. All of these constructions were found by looking at certain functors that were natural from the categorical perspective. The positive part quantum groups admit an action of a "q-boson algebra" with generators E_i and F_i. The E_i act by multiplication by E_i, and the F_i act by the adjoint under a certain (Lusztig's) bilinear form. Adjoints and bilinear forms have natural categorical interpretations. Composing the various F_i in the appropriate way gives the action of the affine generator E_0 in the affine Borel quantum group. Adjoints and bilinear forms have natural categorical interpretations, and carefully studying the corresponding functors led to the results in this paper. Future work in this direction includes extending to other types and in categorifying the prefundamentals directly. This would be especially interesting, since no algebraic categorification of Drinfeld's loop generators is known.

My current project, and what I will be focused on for the rest of my thesis, is in producing new categorifications of several algebras related to quantum groups. The first of these (almost finished!) are the q-boson algebras mentioned above. These categories are defined similarly to Khovanov-Lauda's diagrammatic categorification of the full quantum group. The generating morphisms in this category are like oriented braid diagrams in which we allow upwards braids to turn 180 degrees clockwise, and we allow downwards braids to turn 180 degrees counterclockwise. The main difference now is that the E_i and F_i are only one-sided adjoints, i.e., that the other rotations are not allowed. This new condition adds several useful finiteness properties to the relevant properties makes certain algebraic constructions, like infinite direct sums or localization, much more feasible. It is interesting that this construction works uniformly well in all symmetrizable Kac-Moody types, whereas results for the full quantum group are often restricted to symemtric or even finite types. Currently, I am studying the category obtained by adding infinitely many adjoints to E_i on both sides. I expect that this is related to the deformed Grothendieck group of the Hernandez-Leclerc subcategory of representations of quantum Loop algebras, also called the bosonic extension. The end goal being to provide new structure with which to study characters of quantum loop algebra representations.