Related papers: The Triplet Vertex Operator Algebra W(p) and the R…
Given a vertex operator algebra $V$, one can construct two associative algebras, the Zhu algebra $A(V)$ and the $C_2$-algebra $R(V)$. This gives rise to two abelian categories $A(V)-\text{Mod}$ and $R(V)-\text{Mod}$, in addition to the…
Prototypical rational vertex operator algebras are associated to affine Lie algebras at positive integer level k. They correspond physically to the Wess-Zumino-Witten theories, and their representation theory can be captured by quantum…
In this article the simple modules over the rank-two quantized Weyl algebras at roots of unity over an algebraically closed field are classified.
For coprime $p,q\in\mathbb{Z}_{\geq 2}$, the triplet vertex operator algebra $W_{p,q}$ is a non-simple extension of the universal Virasoro vertex operator algebra of central charge $c_{p,q}=1-\frac{6(p-q)^2}{pq}$, and it is a basic example…
We give a new factorisable ribbon quasi-Hopf algebra U, whose underlying algebra is that of the restricted quantum group for sl(2) at a 2p'th root of unity. The representation category of U is conjecturally ribbon-equivalent to that of the…
A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach, we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the…
We introduce a new family of C_2-cofinite N=1 vertex operator superalgebras SW(m), $m \geq 1$, which are natural super analogs of the triplet vertex algebra family W(p), $p \geq 2$, important in logarithmic conformal field theory. We…
We provide a ribbon tensor equivalence between the representation category of small quantum SL(2), at parameter q=exp($\pi$ i/p), and the representation category of the triplet vertex operator algebra at integral parameter p>1. We provide…
Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolled restricted quantum groups $\overline{U}_q^H(\mathfrak{g})$ at arbitrary roots of unity with a focus on its…
We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal…
To study the representation category of the triplet W-algebra W(p) that is the symmetry of the (1,p) logarithmic conformal field theory model, we propose the equivalent category C(p) of finite-dimensional representations of the restricted…
We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group $\overline{U}_q^H(\mfg)$ of a simple Lie algebra $\mfg$ at roots of unity, and study their categories…
In this paper, we prove the equivalence between two ribbon tensor categories. On the one hand, we consider the category of modules of the Virasoro vertex operator algebra with generic central charge (generic Virasoro VOA) generated by those…
An elementary proof is given for the existence of infinite dimensional abelian subalgebras in quantum W-algebras. In suitable realizations these subalgebras define the conserved charges of various quantum integrable systems. We consider all…
This article undertakes an exploration of simple modules of 3-cyclic quantum Weyl algebra at roots of unity. Under the roots of unity assumption, the algebra becomes a Polynomial Identity algebra and the vector space dimension of the simple…
In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the…
We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the $W$ algebra defined using nilpotent orbit with partition $[q^m,1^s]$. Gauging above AD…
We study the triplet vertex operator algebra $\mathcal{W}(p)$ of central charge $1-\frac{6(p-1)^2}{p}$, $p \geq 2$. We show that $\trip$ is $C_2$-cofinite but irrational since it admits indecomposable and logarithmic modules. Furthermore,…
Galilean $W_3$ vertex operator algebra $\mathcal GW_3(c_L,c_M)$ is constructed as a universal enveloping vertex algebra of certain non-linear Lie conformal algebra. It is proved that this algebra is simple by using determinant formula of…
The notion of the genus of a quadratic form is generalized to vertex operator algebras. We define it as the modular braided tensor category associated to a suitable vertex operator algebra together with the central charge. Statements…