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Cuipo Jiang

Publications and source records attributed to Cuipo Jiang.

At least 19 recordsLinked to original sources
arXiv2025

On singular vectors of simply-laced universal affine vertex operator algebras

Given a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ and a complex number $\kappa$, let $V^{\kappa}(\mathfrak{g})$ be the associated universal affine vertex algebra. Gorelik and Kac [GK07] gave a sufficient and necessary condition for $V^{\kappa}(\mathfrak{g})$ to be simple. In this paper, for simply-laced $\mathfrak{g}$ and non-critical $\kappa$, we determine the weights of singular vectors of $V^{\kappa}(\mathfrak{g})$ with minimal conformal weights, when $V^{\kappa}(\mathfrak{g})$ is not simple. We further determine all the longest Weyl elements in the Kashiwara-Tanisaki character theorem [KT00] which correspond to the weights of the singular vectors.

math.QA↗
arXiv2025

Structure and representations of the coset vertex operator algebra $C( L_{\widehat{osp(1|2)}}(2,0), L_{\widehat{osp(1|2)}}(1,0)^{\otimes 2})$

In this paper, we determinate the structure of the coset vertex operator algebra $C( L_{\widehat{osp(1|2)}}(2,0), L_{\widehat{osp(1|2)}}(1,0)^{\otimes 2})$. We prove that $C( L_{\widehat{osp(1|2)}}(2,0), L_{\widehat{osp(1|2)}}(1,0)^{\otimes 2})$ is an extension of the rational vertex operator algebra $L(c_{10,7},0)$. Representations and fusion rules for $C( L_{\widehat{osp(1|2)}}(2,0), L_{\widehat{osp(1|2)}}(1,0)^{\otimes 2})$ are also completely determined.

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arXiv2024

Associated varieties of simple affine VOAs $L_k(sl_3)$ and $W$-algebras $W_k(sl_3,f)$

In this paper we first prove that the maximal ideal of the universal affine vertex operator algebra $V^k(sl_n)$ for $k=-n+\frac{n-1}{q}$ is generated by two singular vectors of conformal weight $3q$ if $n=3$, and by one singular vector of conformal weight $2q$ if $n\geq 4$. We next determine the associated varieties of the simple vertex operator algebras $L_k(sl_3)$ for all the non-admissible levels $k=-3+\frac{2}{2m+1}$, $m\geq 0$. The varieties of the associated simple affine $W$-algebras $W_k(sl_3,f)$, for nilpotent elements $f$ of $sl_3$, are also determined.

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arXiv2023

Differential graded vertex Lie algebras

This is the continuation of the study of differential graded (dg) vertex algebras previously defined by the authors. The goal of this paper is to construct a functor from the category of dg vertex Lie algebras to the category of dg vertex algebras which is left adjoint to the forgetful functor. This functor not only provides an abundant number of examples of dg vertex algebras, but it is also an important step in constructing a homotopy theory in the category of vertex algebras. Vertex Lie algebras were introduced as analogues of vertex algebras, but in which we only consider the singular part of the vertex operator map and the equalities it satisfies. In this paper, we extend the definition of vertex Lie algebras to the dg setting. We construct a pair of adjoint functors between the categories of dg vertex algebras and dg vertex Lie algebras, which leads to the explicit construction of dg vertex (operator) algebras. We will give examples based on the Virasoro algebra, the Neveu-Schwarz algebra, and dg Lie algebras.

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arXiv2023

Representations of the orbifold of parafermion vertex operator algebra $K(osp(1|2),k)$

This paper is about the orbifold theory of parafermion vertex operator algebras $K(osp(1|2),k)$ associated to the affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(k,0)$ with any positive integer $k$. Among the main results, we classify the irreducible modules for the orbifold of parafermion vertex operator algebra $K(osp(1|2),k)$.

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arXiv2023

Yoneda algebras of the triplet vertex operator algebra

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 category of admissible modules of $V$. In case $V$ is rational and $C_2$-cofinite, the category of admissible $V$-modules and the category of all $A(V)$-modules are equivalent. However, when $V$ is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra $\mathcal{W}(p)$, as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra $ \text{End} (\oplus_{L\in \text{Irr}} P_L)^{op}$, and the Yoneda algebra $\text{Ext}^{*}(\oplus_{L\in \text{Irr}}L, \oplus_{L\in \text{Irr}}L)$. As a consequence, the category of admissible log-modules for the triplet VOA $ \mathcal W(p)$ has infinite global dimension, as do the Zhu algebra $A(\mathcal W(p))$, and the associated graded algebra $\text{gr} \ A(\mathcal W(p))$ which is isomorphic to $R(\mathcal W(p))$. We also describe the Koszul properties of the module categories of $ \mathcal W(p)$, $A(\mathcal W(p))$ and $\text{gr} \ A(\mathcal W(p))$.

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arXiv2022

Cohomological varieties associated to vertex operator algebras

Given a vertex operator algebra V , one can attach a graded Poisson algebra called the C2-algebra R(V). The associate Poisson scheme provides an important invariant for V and has been studied by Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C2-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R(V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted Lie algebras.

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arXiv2021

Tensor Categories arising from the Virasoro Algebra

We show that there is a braided tensor category structure on the category of $C_1$-cofinite modules for the (universal or simple) Virasoro vertex operator algebras of arbitrary central charge. In the generic case of central charge $c=13-6(t+t^{-1})$, with $t \notin \mathbb{Q}$, we prove semisimplicity, rigidity and non-degeneracy and also compute the fusion rules of this tensor category.

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arXiv2020

Simplicity of vacuum modules and associated varieties

In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $\mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $\mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.

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arXiv2019

Representations of the orbifold VOAS $L_{\hat{\frak{sl}_2}}(k,0)^K$ and the commutant VOAS $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)})$

For the Klein group $K$, $k\in\mathbb{Z}_{\geqslant 1}$ and $m\in\mathbb{Z}_{\geqslant 4}$, we study the representations of the orbifold vertex operator algebra $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and the commutant vertex operator algebra of $L_{\hat{\mathfrak{so}_m}}(3,0)$ in $L_{\hat{\mathfrak{so}_m}}(1,0)^{\otimes 3}$ which can be realized as the orbifold vertex operator subalgebra $L_{\hat{\mathfrak{sl}_2}}(2m,0)^{K}$ or its extension. All the irreducible modules for $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)})$ are classified and constructed explicitly.

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arXiv2019

Level-Rank Duality for Vertex Operator Algebras of types B and D

For the simple Lie algebra $ \frak{so}_m$, we study the commutant vertex operator algebra of $ L_{\hat{\frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $ L_{\hat{\frak{so}}_{m}}(1,0)^{\otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{\hat{\frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.

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arXiv2019

Whittaker modules for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[τ]$

The Whittaker module $M_ψ$ and its quotient Whittaker module $L_{ψ, ξ}$ for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[τ]$ are studied. For nonsingular type, it is proved that if $ξ\neq 0$, then $L_{ψ,ξ}$ is irreducible and any irreducible Whittaker $\widehat{H}_{4}[τ]$-module of type $ψ$ with ${\bf k}$ acting as a non-zero scalar $ξ$ is isomorphic to $L_{ψ,ξ}$. Furthermore, for $ξ=0$, all Whittaker vectors of $L_{ψ, 0}$ are completely determined. For singular type, the Whittaker vectors of $L_{ψ, ξ}$ with $ξ\neq 0$ are fully characterized.

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arXiv2019

Fusion rules for $\mathbb{Z}_{2}$-orbifolds of affine and parafermion vertex operator algebras

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra $K(sl_2,k)$ associated to the integrable highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ is the building block of the general parafermion vertex operator $K(\mathfrak{g},k)$ for any finite dimensional simple Lie algebra $\mathfrak{g}$ and any positive integer $k$. We first classify the irreducible modules of $\mathbb{Z}_{2}$-orbifold of the simple affine vertex operator algebra of type $A_1^{(1)}$ and determine their fusion rules. Then we study the representations of the $\mathbb{Z}_{2}$-orbifold of the parafermion vertex operator algebra $K(sl_2,k)$, we give the quantum dimensions, and more technically, fusion rules for the $\mathbb{Z}_{2}$-orbifold of the parafermion vertex operator algebra $K(sl_2,k)$ are completely determined.

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