Research Interests

My research can best be classified with in the areas of Representation Theory (for example, of Braided bialgebras, monoidal categories or 2-categories) and Quantum Algebra (Hopf algebras, Nichols algebras, etc).

Part of my recent work relates to categorification at roots of unity (joint work with Vanessa Miemietz and You Qi). My work with Vladimir Retakh falls into the area of noncommutative Geometry.

My preprints are available on the Arxiv.

Recent Preprints

(with Chelsea Walton) Braided Commutative Algebras over Quantized Enveloping Algebras
Arxiv e-Preprint, January 2019.

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov's full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers ZB(C) from algebras in B-augmented monoidal categories C, where such augmented monoidal categories and the relative monoidal center are defined by the first author in previous work (see this preprint). Here, B is an arbitrary braided monoidal category; Davydov's (and previous works of others) take place in the special case when B is the category of vector spaces Vectk over a field k. Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in B-augmented monoidal categories. Moreover, for a large class of B-augmented monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in Vectk serve as Morita invariants. Many examples are provided throughout.

(with Johannes Flake) On the Monoidal Center of Deligne's Category Rep(St)
Arxiv e-Preprint, January 2019.

We explicitly compute a monoidal subcategory of the monoidal center of Deligne's interpolation category Rep(St) and show that this subcategory is a ribbon category. For t=n a natural number, there exists a functor onto the braided monoidal category of modules over the Drinfeld double of Sn which is essentially surjective and full. Hence the new ribbon category interpolates the categories of crossed modules over the symmetric groups. As an applications, we obtain invariants of framed ribbon links which are polynomials in t. These polynomials interpolate untwisted Dijkgraaf–Witten invariants of the symmetric groups.

Here are slides from a talk presented at this year's Joint Mathematics Meeting in Baltimore, as part of the special session Hopf Algebras and Tensor Categories.

(with You Qi) A Categorification of Cyclotomic Rings
Arxiv e-Preprint June 2018 (v3).

In this paper, we categorify the ring of cyclotomic integers for any order of the root of unity. This construction gives a solution to a question raised by Khovanov (2005). The construction uses a suitable Hopf algebra in the category of q-vector spaces, and a localization of its stable category of modules.
Here are slides from a talk given at the Boston Eastern Sectional Meeting, Spring 2018, about this paper.
Here is a poster presented at the 2018 MAAGC meeting in Philadelphia, about this paper.

The Relative Monoidal Center and Tensor Products of Monoidal Categories
Arxiv e-Preprint April 2018 (v2).

This paper defines a relative version of the monoidal center construction, which is tailored to working with quantum groups. This center construction directly returns a suitable braided monoidal category of quantum group representations as the relative center of modules over its positive part. Moreover, categorical representations of this relative center are studied. For this, the relative tensor product of two monoidal categories that live over the same braided monoidal category is introduced.
Here are slides from a talk given at the conference Recent developments in noncommutative algebra and related areas (Seattle, March 2018), about this project,

(with Vanessa Miemietz) Cell 2-Representations and Categorification at Prime Roots of Unity
Arxiv e-Preprint March 2018 (v2).

A series of papers by V. Mazorchuk and V. Miemietz, and later continued with other collaborators, develops a systematic theory of representations of 2-categories which satisfy natural finiteness assumptions. This theory does not impose or imply conditions of semisimplicity, and is therefore suitable to be applied to known categorifications. In the categorification at prime roots of unity (in the work of M. Khovanov, Y. Qi and B. Elias), one works with categorifications enriched with p-differentials, and takes the stable category of compact modules. This paper extends part of the theory of cell 2-representations to the setup where the 2-categories are enriched with p-differentials. This provides a setup applicable to studying 2-representations in the world of categorification at prime roots of unity.
Here are slides from a talk given at the Riverside Western Sectional Meeting, Fall 2017, about this paper.
Here is a poster presented at the workshop Categorification and Higher Representation Theory at the Mittag-Leffler Institute, about this paper.

List of Publications

(with Vladimir Retakh) Noncommutative Shifted Symmetric Functions
To appear in Moscow Math. J.
Arxiv e-Preprint Arxiv:1807.09197

This paper develops a theory of noncommutative shifted symmetric functions. We introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutative symmetric functions. Shifted versions of ribbon Schur functions are defined and form a basis for the ring. Further, we produce analogues of Jacobi–Trudi and Nägelsbach–Kostka formulas, a duality anti-algebra isomorphism, shifted quasi-Schur functions, and Giambelli's formula in this setup. In addition, an analogue of power sums is provided, satisfying versions of Wronski and Newton formulas. Finally, a realization of these shifted noncommutative symmetric functions as rational functions in noncommuting variables is given. These realizations have a shifted symmetry under exchange of the variables, and are well-behaved under extension of the list of variables
Here is a video of a talk about a preliminary version of this research at V. Retakh's birthday conference. Retakhfest.

Comodule Algebras and Cocycles over the (Braided) Drinfeld Double
Commun. Contemp. Math (online ready), August 2018.

This paper contains the construction of comodule algebras over braided Drinfeld doubles (in particular, over quantized enveloping algebras). Such comodule algebras can be constructed either from comodule algebras or module algebras over one part of the quantum group, in its triangular decomposition.

(with Vladimir Retakh) Algebras of Quasi-Plücker Coordinates are Koszul
J. Pure Appl. Algebra 222 (2018), no. 9, 2810–2822.

In this paper, an abstract algebra of quasi-Pluecker coordinates is consider and shown to be an example of a non-homogeneous Koszul algebra.
Here are slides from a talk given at the Portland Western Sectional Meeting, Spring 2018, about this project,

On Fomin–Kirillov Algebras for Complex Reflection Groups
Comm. Alg. (to appear), September 2016.

Braided Hopf Algebras, Double Constructions, and Applications
DPhil Thesis, University of Oxford, August 2015.

Pointed Hopf Algebras with Triangular Decomposition — A Characterization of Multiparameter Quantum Groups
Algebras and Representation Theory 19 (2016), issue 3, 547–578.
Paper (open access)

Braided Drinfeld and Heisenberg doubles
J. Pure Appl. Algebra 219 (2015), no. 10, 4541–4596.

Last modified on April 15, 2019.