Further results on coset representative categories

Schur-Weyl Duality for Heisenberg Cosets

Let $V$ be a simple vertex operator algebra containing a rank $n$ Heisenberg vertex algebra $H$ and let $C=\text{Com}\left( {H}, {V}\right)$ be the coset of ${H}$ in ${V}$. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of ${C}$ are found and every simple ${C}$-module is shown to be contained in at least one ${V}$-module. A corollary of this is that if ${V}$ is rational and $C_2$-cofi…

Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t = \infty$

Let $λ\in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_λ$, where $W_λ$ is the stabilizer of $λ$ in a finite Weyl group $W$. In this paper, we give a module $\mathbb{K}_{w}(λ)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w λ}(q,\,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $\mat…

Some properties of group-theoretical categories

We first show that every group-theoretical category is graded by a certain double coset ring. As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent. We then give an explicit description of the simple objects in a group-theoretical category (following Ostrik, arXiv:math/0202130) and of the group of invertible objects of a group-theoretical category, in group-theoretical terms. Finally, under certain restrictive conditions, we describe the universal grading group of a group-theoretical category.

Localizations for construction of quantum coset spaces

Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localization…

群论融合范畴中的代数结构

Algebraic structures in group-theoretical fusion categories

本文推广Ostrik和Natale的结果，在群论融合范畴中构造不可分解可分代数的显式Morita等价类代表，并证明其具有Frobenius代数结构及良好性质。

Bar categories and star operations

We introduce the notion of `bar category' by which we mean a monoidal category equipped with additional structure formalising the notion of complex conjugation. Examples of our theory include bimodules over a $*$-algebra, modules over a conventional $*$-Hopf algebra and modules over a more general object which call a `quasi-$*$-Hopf algebra' and for which examples include the standard quantum groups $u_q(g)$ at $q$ a root of unity (these are well-known not to be a usual $*$-Hopf algebra). We also provide examples of strictly quasiassociative bar categories, including modules over `$*$-quasiHop…

模范畴的Grothendieck环：局部化、分层与根

Grothendieck Rings of Module Categories: Localizations, Strata, and Radicals

本文研究非退化辫子融合范畴的模范畴的Grothendieck环，通过定义Drinfeld中心中étale代数的卷积和全中心的同构，建立了环的上三角分层结构，并利用转移矩阵和Gram矩阵计算了多个例子中的层环与根，揭示了非交换性和非零根的存在。

A Higher Frobenius-Schur Indicator Formula for Group-Theoretical Fusion Categories

Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $ω$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the restriction of $ω$. The objects of the category are $G$-graded vector spaces with suitably twisted $H$-actions; the associativity of tensor products is controlled by $ω$. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in $H$ of right $H$-cosets in $G$, with respect to two-cocycles defined by the …

Direct limit completions of vertex tensor categories

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$ super Virasoro algebra to categories of Virasoro algebra modules via certain …

An in-Depth Look at Quotient Modules

The coset $G$-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization $Q$ to Hopf subalgebras, especially the tensor powers and similarity as modules over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism algebras. We prove that $Q$ has a nonzero integral if and only if the modular function restricts to the modular function of the Hopf subalgebra. We also study and organize knowledge of $Q$ and its tensor pow…

可逆融合范畴

Invertible Fusion Categories

本文研究一般域上可逆多重融合范畴的分类，证明其 Morita 等价类由绝对 Galois 群的第三个 Galois 上同调 H^3(K;G_m) 分类，并构造了每个类的融合范畴代表元，同时揭示 Drinfeld 中心辫子等价不蕴含 Morita 等价。

Representation theory of $L_k\left(\mathfrak{osp}(1 | 2)\right)$ from vertex tensor categories and Jacobi forms

The purpose of this work is to illustrate in a family of interesting examples how to study the representation theory of vertex operator superalgebras by combining the theory of vertex algebra extensions and modular forms. Let $L_k\left(\mathfrak{osp}(1 | 2)\right)$ be the simple affine vertex operator superalgebra of $\mathfrak{osp}(1|2)$ at an admissible level $k$. We use a Jacobi form decomposition to see that this is a vertex operator superalgebra extension of $L_k(\mathfrak{sl}_2)\otimes \text{Vir}(p, (p+p')/2)$ where $k+3/2=p/(2p')$ and $\text{Vir}(u, v)$ denotes the regular Virasoro vert…

顶点算子超代数扩张的张量范畴

Tensor categories for vertex operator superalgebra extensions

本文利用张量范畴证明局部代数模与扩张顶点算子超代数模的范畴同构保持辫子幺半结构，并据此推导正则顶点算子超代数的 Verlinde 公式及分类仿费米子陪集的单模、融合规则与模特征变换。