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Zongzhu Lin

Publications and source records attributed to Zongzhu Lin.

At least 19 recordsLinked to original sources
arXiv2025

Differential graded algebras with divided powers and homotopy Lie algebras

Given a commutative algebra $A$ and a quotient $A$-algebra $A/I$, we construct a resolution of $A/I$ as an $A$-module such that it is also a differential graded (dg) algebra with divided powers (PD). This construction makes use of symmetric tensors in the symmetric tensor category of dg $A$-modules and does not require a Noetherian assumption on $A$. Moreover, the resolution has many lifting properties which we leverage to study the homotopy Lie algebra associated to the pair $(A,A/I)$, which is defined as the image in the Yoneda algebra $\operatorname{Ext}^*_{A}(A/I,A/I)$ of the cohomology of the PD derivations of this PD dg algebra. Finally we investigate the complete intersection case in more details as well as connect it to the finite generation of the Yoneda algebra.

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arXiv2025

Equivariant vertex coalgebras, $C_2$-coalgebras and duality for diagonalisable group schemes

In this paper, we define vertex algebras and vertex coalgebras in the category of rational $G_\Gamma$-modules, where $G_\Gamma$ is the group scheme defined by the group algebra $\mathsf k \Gamma$ for an abelian group $\Gamma$. In this context, we introduce the notion of $C_2$-coalgebra for a vertex coalgebra. We prove that there exists a duality between vertex algebras and vertex coalgebras in the category of $G_\Gamma$-modules, and this duality establishes a connection between $C_2$-algebras and $C_2$-coalgebras. Moreover, we also investigate the relationship between their respective modules/comodules.

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arXiv2024

Schur-Weyl Duality for Toroidal Algebras of Type $A$

We state and prove an analog of the Schur-Weyl duality for a quotient of the classical $2$-toroidal Lie algebra of type $A$. We then provide a method to extend this duality to the $m$-toroidal case, $m > 2$.

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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

Realizing Rings of Regular Functions via the Cohomology of Quantum Groups

Let $G$ be a complex reductive group and $P$ be a parabolic subgroup of $G$. In this paper the authors address questions involving the realization of the $G$-module of the global sections of the (twisted) cotangent bundle over the flag variety $G/P$ via the cohomology of the small quantum group. Our main results generalize the important computation of the cohomology ring for the small quantum group by Ginzburg and Kumar, and provides a generalization of well-known calculations by Kumar, Lauritzen, and Thomsen to the quantum case and the parabolic setting. As an application we answer the question (first posed by Friedlander and Parshall for Frobenius kernels) about the realization of coordinate rings of Richardson orbit closures for complex semisimple groups via quantum group cohomology. Formulas will be provided which relate the multiplicities of simple $G$-modules in the global sections with the dimensions of extension groups over the large quantum group.

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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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arXiv2019

Representations and Modules of Rota-Baxter Algebras

We give a broad study of representation and module theory of Rota-Baxter algebras. Regular-singular decompositions of Rota-Baxter algebras and Rota-Baxter modules are obtained under the condition of quasi-idempotency. Representations of an Rota-Baxter algebra are shown to be equivalent to the representations of the ring of Rota-Baxter operators whose categorical properties are obtained and explicit constructions are provided. Representations from coalgebras are investigated and their algebraic Birkhoff factorization is given. Representations of Rota-Baxter algebras in the tensor category context is also formulated.

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arXiv2018

Moduli spaces of conformal structures on Heisenberg vertex algebras

This paper is a continuation to understand Heisenberg vertex algebras in terms of moduli spaces of their conformal structures. We study the moduli space of the conformal structures on a Heisenberg vertex algebra that have the standard fixed conformal gradation. As we know in Proposition 3.1 in Sect.3, conformal vectors of the Heisenberg vertex algebra $V_{\hatη}(1,0)$ that have the standard fixed conformal gradation is parameterized by a complex vector $h$ of its weight-one subspace. First, we classify all such conformal structures of the Heisenberg vertex algebra $V_{\hatη}(1,0)$ by describing the automorphism group of the Heisenberg vertex algebra $V_{\hatη}(1,0)$ and then we describe moduli spaces of their conformal structures that have the standard fixed conformal gradation. Moreover, we study the moduli spaces of semi-conformal vertex operator subalgebras of each of such conformal structures of the Heisenberg vertex algebra $V_{\hatη}(1,0)$. In such cases, we describe their semi-conformal vectors as pairs consisting of regular subspaces and the projections of $h$ in these regular subspaces. Then by automorphism groups $G$ of Heisenberg vertex operator algebras, we get all $G$-orbits of varieties consisting of semi-conformal vectors of these vertex operator algebras. Finally, using properties of these varieties, we give two characterizations of Heisenberg vertex operator algebras.

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arXiv2017

The varieties of semi-conformal vectors of affine vertex operator algebras

This is a continuation of our work to understand vertex operator algebras using the geometric properties of varieties attached to vertex operator algebras. For a class of vertex operator algebras including affine vertex operator algebras associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, we describe their varieties of semi-conformal vectors by some matrix equations. These matrix equations are too complicated to be solved for us. However, for affine vertex operator algebras associated to the simple Lie algebra $\mathfrak{g}$, we find the adjoint group $G$ of $\mathfrak{g}$ acts on the corresponding varieties by a natural way, which implies that such varieties should be described more clearly by studying the corresponding $G$-orbit structures. Based on above methods for general cases, as an example, considering affine vertex operator algebras associated to the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, we shall give the decompositions of $G$-orbits of varieties of their semi-conformal vectors according to different levels. Our results imply that such orbit structures depends on the levels of affine vertex operator algebras associated to a finite dimensional simple Lie algebra $\mathfrak{g}$

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arXiv2017

Geometric Schur Duality of two parameter quantum group of type A

In this paper, we give an geometric description of the Schur-Weyl duality for two-parameter quantum algebras $U_{v, t}(gl_n)$, where $U_{v, t}(gl_n)$ is the deformation of $U_v(I, \cdot)$, the classic Shur-Weyl duality $(U_{r, s}(gl_n), V^{\otimes d}, H_d(r, s))$ can be seen as a corollary of the Shur-Weyl duality $(U_{v, t}(gl_n), V^{\otimes d}, H_d(v, t))$ by using the galois descend approach. we also establish the Shur-Weyl duality between the algebras $\widetilde{U_{v, t}(gl_N)^m}$, $\widehat{U_{v, t}(gl_N)^m}$ and Heck algebra $H_k(v, t)$.

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arXiv2016

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra $V$ with conformal vector $ω$, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and semi-conformal vectors of $(V,ω)$, respectively, and were used to study duality theory of vertex operator algebras via coset constructions. Using these objects attached to $(V,ω)$, we shall understand the structure of the vertex operator algebra $(V,ω)$. At first, we define the set $\on{Sc}(V,ω)$ of semi-conformal vectors of $V$, then we prove that $\on{Sc}(V,ω)$ is an affine algebraic variety with a partial ordering and an involution map. Corresponding to each semi-conformal vector, there is a unique maximal semi-conformal vertex operator subalgebra containing it. The properties of these subalgebras are invariants of vertex operator algebras. As an example, we describe the corresponding varieties of semi-conformal vectors for Heisenberg vertex operator algebras. As an application, we give two characterizations of Heisenberg vertex operator algebras using the properties of these varieties.

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arXiv2016

Representations of Rota-Baxter algebras and regular singular decompositions

There is a Rota-Baxter algebra structure on the field $A=\mathbf{k}((t))$ with $ P$ being the projection map $A=\mathbf{k}[[t]]\oplus t^{-1}\mathbf{k}[t^{-1}]$ onto $ \mathbf{k}[[ t]]$. We study the representation theory and regular-singular decompositions of any finite dimensional $A$-vector space. The main result shows that the category of finite dimensional representations is semisimple and consists of exactly three isomorphism classes of irreducible representations which are all one-dimensional. As a consequence, the number of $GL_A(V)$-orbits in the set of all regular-singular decompositions of an $n$-dimensional $ A$-vector space $V$ is $(n+2)(n+1)/2$. We also use the result to compute the generalized class number, i.e., the number of the $GL_n(A)$-isomorphism classes of finitely generated $\mathbf{k}[[t]]$-submodules of $A^n$.

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arXiv2014

Tensor Decomposition, Parafermions, Level-Rank Duality, and Reciprocity Law for Vertex Operator Algebras

For the semisimple Lie algebra $ \frak{sl}_n$, the basic representation $L_{\widehat{\frak{sl}_{n}}}(1,0)$ of the affine Lie algebra $\widehat{\frak{sl}_{n}}$ is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of $ L_{\widehat{\frak{sl}_{n}}}(l,0)$ in the $l$-fold tensor product $ L_{\widehat{\frak{sl}_{n}}}(1,0)^{\otimes l}$ is isomorphic to the parafermion vertex operator algebra $K(\frak{sl}_{l},n)$, which is the commutant of the Heisenberg vertex operator algebra $L_{\widehat{\frak{h}}}(n,0) $ in $L_{\widehat{\frak{sl}_l}}(n,0)$. The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of $ L_{\widehat{\frak{sl}_{n}}}(l_1+\cdots +l_s, 0)$ in $L_{\widehat{\frak{sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat{\frak{sl}_{n}}}(l_s, 0)$ is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of $\frak{sl}_{l_1+\cdots+l_s}$ corresponding to the composition $(l_1, \cdots, l_s)$ in the rational vertex operator algebra $ L_{\widehat{\frak{sl}}_{l_1+\cdots +l_s}}(n,0)$. This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras.

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arXiv2014

Quiver representations of constant Jordan type and vector bundles

Inspired by the work of Benson, Carlson, Friedlander, Pevtsova, and Suslin on modules of constant Jordan type for finite group schemes, we introduce in this paper the class of representations of constant Jordan type for an acyclic quiver $Q$. We do this by first assigning to an arbitrary finite-dimensional representation of $Q$ a sequence of coherent sheaves on moduli spaces of thin representations. Next, we show that our quiver representations of constant Jordan type are precisely those representations for which the corresponding sheaves are locally free. We also construct representations of constant Jordan type with desirable homological properties. Finally, we show that any element of $\mathbb{Z}^L$, where $L$ is the Loewy length of the path algebra of $Q$, can be realized as the Jordan type of a virtual representation of $Q$ of relative constant Jordan type.

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arXiv2013

The commutant of $L_{\widehat{\frak{sl}}_{2}}(n,0)$ in the vertex operator algebra $L_{\widehat{\frak{sl}}_{2}}(1,0)^{\otimes n}$

We study the commutant $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$ of $L_{\widehat{\frak{sl}}_{2}}(n,0)$ in the vertex operator algebra $L_{\widehat{\frak{sl}}_{2}}(1,0)^{\otimes n}$, for $n\geq 2$. The main results include a complete classification of all irreducible $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$-modules and a proof that $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$ is a rational vertex operator algebra. As a consequence, every irreducible $L_{\widehat{\frak{sl}}_{2}}(n,0)^c$-module arises from the coset construction as conjectured in \cite{LS}.

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arXiv2013

Approach to artinian algebras via natural quivers

Given an Artinian algebra $A$ over a field $k$, there are several combinatorial objects associated to $A$. They are the diagram $D_A$ as defined in [DK], the natural quiver $Δ_A$ defined in \cite{Li} (cf. Section 2), and a generalized version of $k$-species $(A/r, r/r^2)$ with $r$ being the Jacobson radical of $A$. When $A$ is splitting over the field $k$, the diagram $D_A$ and the well-known ext-quiver $Γ_A$ are the same. The main objective of this paper is to investigate the relations among these combinatorial objects and in turn to use these relations to give a characterization of the algebra $A$.

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