Research

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Quantum groups

• "Categorical approach to dynamical quantum groups" (with Pavel Safronov), arXiv:2008.09081, published

  We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups F(G) and F_q(G) are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group.

• "Geometric and categorical approaches to dynamical quantum groups", PhD thesis, link

  In addition to the previous paper, we present classical versions of the dynamical constructions in terms of shifted Poisson structures. Also, we begin a study of the Whittaker analog of the parabolic restriction functor that resulted in the next paper later.

• "Yangians, mirabolic subalgebras, and Whittaker vectors", arXiv:2310.06669,

  We construct an element, which we call the Kirillov projector, that connects the topics of the title: on the one hand, it is defined using the Yangian of gl_N, on the other hand, it gives a canonical projection onto the space of Whittaker vectors for any Whittaker module over the mirabolic subalgebra. Using the Kirillov projector, we deduce some categorical properties of Whittaker modules, for instance, we prove a mirabolic analog of Skryabin's theorem. We also show that it quantizes a rational version of the Cremmer-Gervais r-matrix. As application, we construct a universal vertex-IRF transformation from the standard dynamical R-matrix to this constant one in categorical terms.

• "Intertwining operators between subregular Whittaker modules for gl_N and non-standard quantizations" (with Brian Li), arXiv:2310.19040

  In this paper, we study intertwining operators between subregular Whittaker modules of $\gl_N$ generalizing, on the one hand, the classical exchange construction of dynamical quantum groups, on the other hand, earlier results for principal W-algebras. We explicitly construct them using the generators of W-algebras introduced by Brundan-Kleshchev. We interpret the fusion on intertwining operators in terms of categorical actions and compute the semi-classical limit of the corresponding monoidal isomorphisms which turn out to depend on dynamical-like parameters.

Other

• "Lectures on Factorization Homology, \infty-Categories, and Topological Field Theories" (Hiro Lee Tanaka, with contribution by Araminta Amabel, me, Lukas Müller) arXiv:1907.00066 , published

  Typesetting lecture notes by Hiro Lee Tanaka at Summer School: Geometric Representation Theory and Low-dimensional Topology, Edinburgh 2019