Related papers: Generating functions for ranks of pre-modular cate…

In this paper we study generating functions resembling the rank of strongly unimodal sequences. We give combinatorial interpretations, identities in terms of mock modular forms, asymptotics, and a parity result. Our functions imitate a…

We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce \emph{Grothendieck unitarizability} as a natural generalization of unitarizability to any class of…

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found,…

For each compact, simple, simply-connected Lie group and each integer level we construct a modular tensor category from a quotient of a certain subcategory of the category of representations of the corresponding quantum group. We determine…

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an…

In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks.

In this paper we present a generating function approach to two counting problems in elementary quantum mechanics. The first is to find the total ways of distributing identical particles among different states. The second is to find the…

A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely…

We develop categorical and number theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank $8$. In particular we find three distinct…

In this paper we study modular tensor categories (braided rigid balanced tensor categories with additional finiteness and non-degeneracy conditions), in particular, representations of quantum groups at roots of unity. We show that the…

We define a new class of integrable vertex models associated to quantum groups at roots of unit

We sketch a simplification of proofs of old results on the arithmeticity of the group generated by opposing integral unipotent radicals in higher rank arithmetic groups

It is known that finite crossed modules provide premodular tensor categories. These categories are in fact modularizable. We construct the modularization and show that it is equivalent to the module category of a finite Drinfeld double.

In this work we define a unified generating functions for 9 different kinds of set partitions including cyclically ordered set partitions. Such generating function depends on 4 parameters. We consider property of this function and provide…

We survey a number of classification tools developed in recent years and employ them to classify pseudo-unitary rank 5 premodular categories up to equivalence.

Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several…

Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those…

We introduce a generating function associated to the homogeneous generators of a graded algebra that measures how far is this algebra from being finitely generated. For the case of some algebras of Frobenius endomorphisms we describe this…

Closed-form generating functions for counting one-face rooted hypermaps with a known number of darts by number of vertices and edges is found, using matrix integral expressions relating to the reduced density operator of a bipartite quantum…

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized $n\times r$ matrices as well as quantized factor algebras of $M_q(n)$ are analyzed. The latter are the quantized function…