In this paper, we first give a 1-1 corresponds between torus C/Λ and cubic curve C in P_C^2. As complex manifold, they are isomorphic, therefore we can treat C/Λ as a variety and construction a vertex algebra sheaf ...In this paper, we first give a 1-1 corresponds between torus C/Λ and cubic curve C in P_C^2. As complex manifold, they are isomorphic, therefore we can treat C/Λ as a variety and construction a vertex algebra sheaf on it.展开更多
For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on t...For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C（（t^1/p））.展开更多
In this paper, we consider an integral basis for affine vertex algebra Vk (sl2) when the level k is integral by a direct calculation, then use the similar way to analyze an integral basis for Virasoro vertex algebra V...In this paper, we consider an integral basis for affine vertex algebra Vk (sl2) when the level k is integral by a direct calculation, then use the similar way to analyze an integral basis for Virasoro vertex algebra Vvir (2k,0). Finally, we take the combination of affine algebras and Virasoro Lie algebras into consideration. By analogy with the construction of Lie algebras over Z using Chevalley bases, we utilize the Z-basis of Lav whose structure constants are integral to find an integral basis for the universal enveloping algebra of it.展开更多
In this paper,we shall study structures of even lattice vertex operator algebras by using the geometry of the varieties of their semi-conformal vectors.We first give the varieties of semi-conformal vectors of a family...In this paper,we shall study structures of even lattice vertex operator algebras by using the geometry of the varieties of their semi-conformal vectors.We first give the varieties of semi-conformal vectors of a family of vertex operator algebras V_(√kA_(1)) associated to rank-one positive definite even lattices √kA_(1) for arbitrary positive integers k to characterize these even lattice vertex operator algebras.In such a family of lattice vertex operator algebras V_(√kA_(1)),the vertex operator algebra V_(√2A_(1)) is different from others.Hence we describe the varieties of semi-conformal vectors of V_(√2A_(1)) and the fixed vertex operator subalgebra V^(+)√2A_(1).Moreover,as applications,we study the relations between vertex operator algebras V_(√kA_(1) )and L_(sl_(2))(k,0)for arbitrary positive integers k by the viewpoint of semi-conformal homomorphisms of vertex operator algebras.For case k=2,in the series of rational simple affine vertex operator algebras L_(sl_(2))(k,0)for positive integers k,we show that L_(sl_(2))(2,0)is a unique frame vertex operator algebra with rank 3.展开更多
We introduce and study the concept of （weak） pseudotwistor for a nonlocal vertex algebra, as a generalization of the notion of twistor. We give the relations between pseudotwistors and twisting operators. Furthermor...We introduce and study the concept of （weak） pseudotwistor for a nonlocal vertex algebra, as a generalization of the notion of twistor. We give the relations between pseudotwistors and twisting operators. Furthermore, we study the inverse of an invertible weak pseudotwistor and the composition of two weak pseudotwistors.展开更多
In this paper, we introduce an abstract method of constructing vertex algebras which satisfy the C1-cofiniteness condition. We give some examples to explain how to use this abstract method.
The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one.In this paper,we first study the variety of semi-conformal vecto...The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one.In this paper,we first study the variety of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra,which is a finite set consisting of two nontrivial elements.Based on this property,we also show that the twisted Heisenberg-Virasoro vertex operator algebra is a tensor product of two vertex operator algebras.Moreover,associating to properties of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra,we charaterized twisted Heisenberg-Virasoro vertex operator algebras.This will be used to understand the classification problems of vertex operator algebras whose varieties of semi-conformal vectors are finite sets.展开更多
We investigate modular framed vertex operator algebras over an algebraically closed field F whose characteristic is different from 2 and 7.In particular,the rationality of modular framed vertex operator algebras is es...We investigate modular framed vertex operator algebras over an algebraically closed field F whose characteristic is different from 2 and 7.In particular,the rationality of modular framed vertex operator algebras is established.For a modular code vertex operator algebra,the irreducible modules are constructed and classified.Moreover,a Z[1/2]-form for any framed vertex operator algebra over C is constructed.As a result,one can obtain a modular framed vertex operator algebra from any framed vertex operator algebra over C.展开更多
We prove a general mirror duality theorem for a subalgebra U of a simple conformal vertex algebra A and its commutant V=ComA(U).Specifically,we assume that A≌■_(i∈I)U_(i)■V_(i) as a U■V-module,where the U-modules...We prove a general mirror duality theorem for a subalgebra U of a simple conformal vertex algebra A and its commutant V=ComA(U).Specifically,we assume that A≌■_(i∈I)U_(i)■V_(i) as a U■V-module,where the U-modules Uiare simple and distinct and are objects of a semisimple braided ribbon category of Umodules,and the V-modules Viare semisimple and contained in a(not necessarily rigid) braided tensor category of V-modules.We also assume U=ComA(V).Under these conditions,we construct a braid-reversed tensor equivalence τ:u_(A)→v_(A),where u_(A)is the semisimple category of U-modules with simple objects Ui,i∈I,and v_(A)is the category of V-modules whose objects are finite direct sums of Vi.In particular,the V-modules Viare simple and distinct,and v_(A)is a rigid tensor category.As an application,we find a rigid semisimple tensor subcategory of modules for the Virasoro algebra at central charge 13+6p+6p^(-1),p∈Z_(≥2), which is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl2-modules.Consequently,the Virasoro vertex operator algebra at central charge 13+6p+6p^(-1)is the PSL_(2)(C)-fixed-point subalgebra of a simple conformal vertex algebra w(-p),analogous to the realization of the Virasoro vertex operator algebra at central charge 13-6p-6p^(-1)as the PSL_(2)(C)-fixed-point subalgebra of the triplet algebra W(p).展开更多
基金Supported by the National Natural Science Foundation of China(11475178,11571119)
文摘In this paper, we first give a 1-1 corresponds between torus C/Λ and cubic curve C in P_C^2. As complex manifold, they are isomorphic, therefore we can treat C/Λ as a variety and construction a vertex algebra sheaf on it.
基金National Key Basic Research Project of China under Grant Nos.2004CB318000 and 2006CB805905National Natural Science Foundation of China under Grant No.10471034+1 种基金the Outstanding Youth Fund of Henan Province under Grant No.0512000100Innovation Fund of Colleges and Universities in Henan Province
文摘For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C（（t^1/p））.
文摘In this paper, we consider an integral basis for affine vertex algebra Vk (sl2) when the level k is integral by a direct calculation, then use the similar way to analyze an integral basis for Virasoro vertex algebra Vvir (2k,0). Finally, we take the combination of affine algebras and Virasoro Lie algebras into consideration. By analogy with the construction of Lie algebras over Z using Chevalley bases, we utilize the Z-basis of Lav whose structure constants are integral to find an integral basis for the universal enveloping algebra of it.
基金Supported by National Natural Science Foundation of China(Grant No.12475002).
文摘In this paper,we shall study structures of even lattice vertex operator algebras by using the geometry of the varieties of their semi-conformal vectors.We first give the varieties of semi-conformal vectors of a family of vertex operator algebras V_(√kA_(1)) associated to rank-one positive definite even lattices √kA_(1) for arbitrary positive integers k to characterize these even lattice vertex operator algebras.In such a family of lattice vertex operator algebras V_(√kA_(1)),the vertex operator algebra V_(√2A_(1)) is different from others.Hence we describe the varieties of semi-conformal vectors of V_(√2A_(1)) and the fixed vertex operator subalgebra V^(+)√2A_(1).Moreover,as applications,we study the relations between vertex operator algebras V_(√kA_(1) )and L_(sl_(2))(k,0)for arbitrary positive integers k by the viewpoint of semi-conformal homomorphisms of vertex operator algebras.For case k=2,in the series of rational simple affine vertex operator algebras L_(sl_(2))(k,0)for positive integers k,we show that L_(sl_(2))(2,0)is a unique frame vertex operator algebra with rank 3.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11201285, 11371238） and a grant of the First-class Discipline of Universities in Shanghai.
文摘We introduce and study the concept of （weak） pseudotwistor for a nonlocal vertex algebra, as a generalization of the notion of twistor. We give the relations between pseudotwistors and twisting operators. Furthermore, we study the inverse of an invertible weak pseudotwistor and the composition of two weak pseudotwistors.
文摘In this paper, we introduce an abstract method of constructing vertex algebras which satisfy the C1-cofiniteness condition. We give some examples to explain how to use this abstract method.
基金supported by The Key Research Project of Institutions of Higher Education in Henan Province,P.R.China(No.17A11003)
文摘The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one.In this paper,we first study the variety of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra,which is a finite set consisting of two nontrivial elements.Based on this property,we also show that the twisted Heisenberg-Virasoro vertex operator algebra is a tensor product of two vertex operator algebras.Moreover,associating to properties of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra,we charaterized twisted Heisenberg-Virasoro vertex operator algebras.This will be used to understand the classification problems of vertex operator algebras whose varieties of semi-conformal vectors are finite sets.
基金supported by the Simons Foundation(Grant No.634104)Ching Hung Lam was supported by Ministry of Science and Technology(Grant No.104-2115-M-001-004-MY3)supported by National Natural Science Foundation of China(Grant No.12071314)。
文摘We investigate modular framed vertex operator algebras over an algebraically closed field F whose characteristic is different from 2 and 7.In particular,the rationality of modular framed vertex operator algebras is established.For a modular code vertex operator algebra,the irreducible modules are constructed and classified.Moreover,a Z[1/2]-form for any framed vertex operator algebra over C is constructed.As a result,one can obtain a modular framed vertex operator algebra from any framed vertex operator algebra over C.
文摘We prove a general mirror duality theorem for a subalgebra U of a simple conformal vertex algebra A and its commutant V=ComA(U).Specifically,we assume that A≌■_(i∈I)U_(i)■V_(i) as a U■V-module,where the U-modules Uiare simple and distinct and are objects of a semisimple braided ribbon category of Umodules,and the V-modules Viare semisimple and contained in a(not necessarily rigid) braided tensor category of V-modules.We also assume U=ComA(V).Under these conditions,we construct a braid-reversed tensor equivalence τ:u_(A)→v_(A),where u_(A)is the semisimple category of U-modules with simple objects Ui,i∈I,and v_(A)is the category of V-modules whose objects are finite direct sums of Vi.In particular,the V-modules Viare simple and distinct,and v_(A)is a rigid tensor category.As an application,we find a rigid semisimple tensor subcategory of modules for the Virasoro algebra at central charge 13+6p+6p^(-1),p∈Z_(≥2), which is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl2-modules.Consequently,the Virasoro vertex operator algebra at central charge 13+6p+6p^(-1)is the PSL_(2)(C)-fixed-point subalgebra of a simple conformal vertex algebra w(-p),analogous to the realization of the Virasoro vertex operator algebra at central charge 13-6p-6p^(-1)as the PSL_(2)(C)-fixed-point subalgebra of the triplet algebra W(p).
基金Supported by the National Natural Science Foundation (10471034) Supported by the Provincial Foundationof Outstanding Young Scholars of Henan (0512000100) Supported by the Foundation of Henan University (05YBZR013)
