Zongzhu Lin
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Zongzhu Lin has a total of 23 co-authors. Here are the top co-authors by number of shared publications:
- Yanjun Chu
School of Mathematics and Statistics, Henan University, Kaifeng, China
5 shared publications - Yiqiang Li
Mathematics Department , Yale University , New Haven, Connecticut, USA
4 shared publications - Yiqiang Li3 shared publications
- Chongying Dong
University of California
2 shared publications - Cuipo Jiang
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P. R. China
2 shared publications - Daniel K. Nakano
Department of Mathematics, University of Georgia, Athens, Georgia 30602
2 shared publications
Latest Publications
- 4 April 2024
- Published by Elsevier in Journal of Geometry and Physics
- Vol. 200
The classification of semi-conformal structures of Heisenberg vertex operator algebras
- 4 April 2024
- Published by Elsevier in Journal of Geometry and Physics
- Vol. 200
The classification of semi-conformal structures of Heisenberg vertex operator algebras
- Open Access
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.Open AccessRealizing 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. - Open Access
Differential graded vertex operator algebras and their Poisson algebras
In this paper, we define differential graded vertex operator algebras and the algebraic structures on the associated Zhu algebras and $C_2$-algebras. We also introduce the corresponding notions of modules, and investigate the relations between the different module categories.
Open AccessDifferential graded vertex operator algebras and their Poisson algebras
In this paper, we define differential graded vertex operator algebras and the algebraic structures on the associated Zhu algebras and $C_2$-algebras. We also introduce the corresponding notions of modules, and investigate the relations between the different module categories.
- 3 October 2022
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 375 (12) , 8325-8352
Tensor decomposition, parafermions, level-rank duality, and reciprocity law for vertex operator algebras
For a semisimple Lie algebra s l n \frak {sl}_n , the basic representation L s l n ^ ( 1 , 0 ) L_{\widehat {\frak {sl}_{n}}}(1,0) of the affine Lie algebra s l n ^ \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 s l n ^ ( l , 0 ) L_{\widehat {\frak {sl}_{n}}}(l,0) in the l l -fold tensor product L s l n ^ ( 1 , 0 ) ⊗ l L_{\widehat {\frak {sl}_{n}}}(1,0)^{\otimes l} is isomorphic to the parafermion vertex operator algebra K ( s l l , n ) K(\frak {sl}_{l},n) , which is the commutant of the Heisenberg vertex operator algebra L h ^ ( n , 0 ) L_{\widehat {\frak {h}}}(n,0) in L s l l ^ ( n , 0 ) 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 s l n ^ ( l 1 + ⋯ + l s , 0 ) L_{\widehat {\frak {sl}_{n}}}(l_1+\cdots +l_s, 0) in L s l n ^ ( l 1 , 0 ) ⊗ ⋯ ⊗ L s l n ^ ( l s ...
- 3 October 2022
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 375 (12) , 8325-8352
Tensor decomposition, parafermions, level-rank duality, and reciprocity law for vertex operator algebras
For a semisimple Lie algebra s l n \frak {sl}_n , the basic representation L s l n ^ ( 1 , 0 ) L_{\widehat {\frak {sl}_{n}}}(1,0) of the affine Lie algebra s l n ^ \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 s l n ^ ( l , 0 ) L_{\widehat {\frak {sl}_{n}}}(l,0) in the l l -fold tensor product L s l n ^ ( 1 , 0 ) ⊗ l L_{\widehat {\frak {sl}_{n}}}(1,0)^{\otimes l} is isomorphic to the parafermion vertex operator algebra K ( s l l , n ) K(\frak {sl}_{l},n) , which is the commutant of the Heisenberg vertex operator algebra L h ^ ( n , 0 ) L_{\widehat {\frak {h}}}(n,0) in L s l l ^ ( n , 0 ) 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 s l n ^ ( l 1 + ⋯ + l s , 0 ) L_{\widehat {\frak {sl}_{n}}}(l_1+\cdots +l_s, 0) in L s l n ^ ( l 1 , 0 ) ⊗ ⋯ ⊗ L s l n ^ ( l s ...
- 1 January 2021
- Published by International Press of Boston in Asian Journal of Mathematics
- Vol. 25 (6) , 841-870
Representations and modules of Rota–Baxter algebras
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2021
- Published by International Press of Boston in Asian Journal of Mathematics
- Vol. 25 (6) , 841-870
Representations and modules of Rota–Baxter algebras
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- Open Access
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.
Open AccessRepresentations 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.
- Open Access
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 $V_{\hat{\eta}}(1,0)$ 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 $V_{\hat{\eta}}(1,0)$ by describing the automorphism group of the Heisenberg vertex algebra $V_{\hat{\eta}}(1,0)$ 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 $V_{\hat{\eta}}(1,0)$. 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.Open AccessModuli 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 $V_{\hat{\eta}}(1,0)$ 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 $V_{\hat{\eta}}(1,0)$ by describing the automorphism group of the Heisenberg vertex algebra $V_{\hat{\eta}}(1,0)$ 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 $V_{\hat{\eta}}(1,0)$. 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. - 1 December 2018
- Published by Elsevier in Journal of Algebra
- Vol. 515, 77-101
The varieties of semi-conformal vectors of affine vertex operator algebras
- 1 December 2018
- Published by Elsevier in Journal of Algebra
- Vol. 515, 77-101
The varieties of semi-conformal vectors of affine vertex operator algebras
- Open Access
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 $\mathfrak{g}$Open AccessThe 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 $\mathfrak{g}$ - Open Access
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 $H_k(v, t)$.Open AccessGeometric 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 $H_k(v, t)$.