顶点算子超代数的表示
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摘要
超对称性在二维共形场论中扮演了重要的作用,这促使我们来研究顶点算子超代数及其表示理论.顶点算子超代数是顶点算子代数的自然推广.H. Tsukada (1990)研究了一些重要的顶点算子超代数.V. G. Kac和W. Wang (1996)详细地研究了三类重要的顶点算子超代数,即关于仿射Kac-Moody超代数、Neveu-Schwarz (NS)代数、自由费密子的顶点算子超代数.他们把Y. Zhu的A(V)-理论推广为顶点算子超代数的A(V)-理论,并讨论了顶点算子超代数的表示理论.H. Li (1996)利用顶点算子的Local system构造并讨论了顶点算子超代数及其表示.徐晓平(1998)在其书中也给出了许多有关顶点算子超代数及其模的知识.本文,我们进一步研究了顶点算子超代数和相关的结合代数的表示,顶点算子超代数的双模理论,以及利用Neveu-Schwarz(NS)-型的顶点算子超代数和二元线性码理论构造了一类编码顶点算子超代数,进而研究了其表示理论.本文将分三部分来分别讨论顶点算子超代数的表示问题.
第一部分：设V是一个顶点算子超代数.我们得到了一系列的结合代数An(V)(对任何n∈i/2+(?)+(i∈{0,1})).我们也给出了An(V)-模但非An-1/2(V)-模的不可约模范畴和单的可容许的V-模的范畴之间的一一对应关系.对于给定的An(V)-模但非An-1/2(V)-模U,我们还构造了一类广义Verma可容许的V-模Mn(U).进而利用结合代数的表示进一步研究了顶点算子超代数的表示论.
第二部分：对于任意一个顶点算子超代数V,m,n∈1/2(?)+,通过构造An(V)-Am(V)-双模An,m(V),讨论了双模An,m(V)的性质.刻画了V的一个从可容许的V-模的第m层子空间到第n层子空间的作用.我们利用An,m(V)和Am(V)-模U还构造了一类Verma型可容许的V-模M(U)=(?)n∈1/2Z+An,m(V)(?)Am(V)U,证明了M(U)与本文第一章构造的广义Verma可容许的V-模M(U)的确是同构关系.
第三部分：我们首先讨论了顶点算子超代数的张量积.进而,我们利用任意一个含有奇重量的二元码研究了编码顶点算子超代数的表示论.利用M. Miyamoto的证明,我们证明了满足本文假设的(V，Y)同构于MD,其中D是某个含有奇重量码字的二元线性码.此外,我们也证明了对任意一个含有奇重量码字的二元线性编码D,编码顶点算子超代数MD是有理的.进而,我们利用M. Miyamoto的结论和诱导模的方法给出了MD的不可约表示的一般形式.利用这个结果,我们进一步研究了汉明顶点算子超代数MH7的表示.证明了在汉明顶点算子超代数MH7中只存在一组7个相互正交的中心电荷为1／2的共形向量.并且我们还给出了所有的汉明顶点算子超代数的不可约表示的分类.
Vertex operator superalgebras can be considered as natural generalizations of vertex operator algebras. The supersymmetry, which plays an important role in two-dimensional conformal field theories, is one of the main reasons to study vertex operator superalgebras. There has been a rapid development in the theory of vertex operator superalgebras over the past decades. Several important types of vertex operator superalgebras have been studied since 1990 by H. Tsukada. The Zhu's A(V)-theory on vertex operator algebras was generalized to A(V)-theoty on vertex operator superalgebras by V. G. Kac and W. Wang in 1996. They also studied in detail three classes of vertex operator superalgebras, i.e., the vertex operator su-peralgebras associated to the affine Kac-Moody superalgebras, the Neveu-Schwarz algebras, and the free fermions. The representations of these vertex operator su-peralgebras were also studied. By the " local system of vertex operators " for a (super) vector space, H. Li proved that any local system of vertex operators on a (super) vector space M has a natural vertex (super)algebra structure with M as a module and studied the representations of vertex operator superalgebras in 1996. For more results on the theory of vertex operator superalgebras and their repre-sentations on can refer to Xu'book pressed in 1998. In this paper, we study the representations of vertex operator superalgebras and related associative algebras, bimodules associated to vertex operator superalgebras and the representations of code vertex operator superalgebras obtained by combining the minimal vertex op-erator superalgebra L(1/2,0)(?)L(1/2,1/2) with a binary linear code which contains codewords of odd weight. The present paper includes three main parts.
In the first part, let V be a vertex operator superalgebra. We construct a sequence of associative algebras An(V) for n∈1/2Z+. It is also exposed that there is a pair of functors between the category of An(V)-modules which are not An-1/2(V)- modules and the category of admissible V-modules. The functors exhibit a bijection between the simple modules in each category. We also construct a generalized Verma admissible V-module Mn(U) from an An(V)-module U which is not an An-1/2(V)-module. Furthermore, we study the theory of representations of vertex operator superalgebras by associative algebras An(V), n∈1/2Z+.
In the second part, let V be a vertex operator superalgebra and m,n∈1/2Z+. We construct an An(V)-Am(V)-bimodule An,m(V) which characterizes the action of V from the level m subspace to level n subspace of an admissible V-module. We study the properties of the An(V)-Am(V)-bimodule An,m(V) and discuss relations between An(V)-modules and admissible V-modules. We also construct a Verma type admissible V-module M(U)=(?)n∈Z An,m(V)(?)Am(v) U, which is proved to be isomorphic to the M(U) defined in the first part of this paper.
In the third part, we study the structure of the tensor product of vertex op-erator superalagebras. Furthermore, we study the representations of code vertex operator superalgebras resulting from a binary linear code which contains code-words of odd weight. We prove that the code vertex operator superalgebra MD is rational, and Miyamoto's construction of induced modules is generalized to code vertex operator superalgebra. We show that there exists only one set of mutually orthogonal seven conformal vectors with central charge 1/2 in the Hamming code vertex operator superalgebra MH7, and we classify all irreducible MH7-modules. As in [29] and [31], our main tool is the theory of induced modules developed by Dong and Lin in [33] and fusion rules of the rational vertex operator algebra L(1/2,0) with central charge 1/2 (see [26]).
第一部分：设V是一个顶点算子超代数.我们得到了一系列的结合代数An(V)(对任何n∈i/2+(?)+(i∈{0,1})).我们也给出了An(V)-模但非An-1/2(V)-模的不可约模范畴和单的可容许的V-模的范畴之间的一一对应关系.对于给定的An(V)-模但非An-1/2(V)-模U,我们还构造了一类广义Verma可容许的V-模Mn(U).进而利用结合代数的表示进一步研究了顶点算子超代数的表示论.
第二部分：对于任意一个顶点算子超代数V,m,n∈1/2(?)+,通过构造An(V)-Am(V)-双模An,m(V),讨论了双模An,m(V)的性质.刻画了V的一个从可容许的V-模的第m层子空间到第n层子空间的作用.我们利用An,m(V)和Am(V)-模U还构造了一类Verma型可容许的V-模M(U)=(?)n∈1/2Z+An,m(V)(?)Am(V)U,证明了M(U)与本文第一章构造的广义Verma可容许的V-模M(U)的确是同构关系.
第三部分：我们首先讨论了顶点算子超代数的张量积.进而,我们利用任意一个含有奇重量的二元码研究了编码顶点算子超代数的表示论.利用M. Miyamoto的证明,我们证明了满足本文假设的(V，Y)同构于MD,其中D是某个含有奇重量码字的二元线性码.此外,我们也证明了对任意一个含有奇重量码字的二元线性编码D,编码顶点算子超代数MD是有理的.进而,我们利用M. Miyamoto的结论和诱导模的方法给出了MD的不可约表示的一般形式.利用这个结果,我们进一步研究了汉明顶点算子超代数MH7的表示.证明了在汉明顶点算子超代数MH7中只存在一组7个相互正交的中心电荷为1／2的共形向量.并且我们还给出了所有的汉明顶点算子超代数的不可约表示的分类.
Vertex operator superalgebras can be considered as natural generalizations of vertex operator algebras. The supersymmetry, which plays an important role in two-dimensional conformal field theories, is one of the main reasons to study vertex operator superalgebras. There has been a rapid development in the theory of vertex operator superalgebras over the past decades. Several important types of vertex operator superalgebras have been studied since 1990 by H. Tsukada. The Zhu's A(V)-theory on vertex operator algebras was generalized to A(V)-theoty on vertex operator superalgebras by V. G. Kac and W. Wang in 1996. They also studied in detail three classes of vertex operator superalgebras, i.e., the vertex operator su-peralgebras associated to the affine Kac-Moody superalgebras, the Neveu-Schwarz algebras, and the free fermions. The representations of these vertex operator su-peralgebras were also studied. By the " local system of vertex operators " for a (super) vector space, H. Li proved that any local system of vertex operators on a (super) vector space M has a natural vertex (super)algebra structure with M as a module and studied the representations of vertex operator superalgebras in 1996. For more results on the theory of vertex operator superalgebras and their repre-sentations on can refer to Xu'book pressed in 1998. In this paper, we study the representations of vertex operator superalgebras and related associative algebras, bimodules associated to vertex operator superalgebras and the representations of code vertex operator superalgebras obtained by combining the minimal vertex op-erator superalgebra L(1/2,0)(?)L(1/2,1/2) with a binary linear code which contains codewords of odd weight. The present paper includes three main parts.
In the first part, let V be a vertex operator superalgebra. We construct a sequence of associative algebras An(V) for n∈1/2Z+. It is also exposed that there is a pair of functors between the category of An(V)-modules which are not An-1/2(V)- modules and the category of admissible V-modules. The functors exhibit a bijection between the simple modules in each category. We also construct a generalized Verma admissible V-module Mn(U) from an An(V)-module U which is not an An-1/2(V)-module. Furthermore, we study the theory of representations of vertex operator superalgebras by associative algebras An(V), n∈1/2Z+.
In the second part, let V be a vertex operator superalgebra and m,n∈1/2Z+. We construct an An(V)-Am(V)-bimodule An,m(V) which characterizes the action of V from the level m subspace to level n subspace of an admissible V-module. We study the properties of the An(V)-Am(V)-bimodule An,m(V) and discuss relations between An(V)-modules and admissible V-modules. We also construct a Verma type admissible V-module M(U)=(?)n∈Z An,m(V)(?)Am(v) U, which is proved to be isomorphic to the M(U) defined in the first part of this paper.
In the third part, we study the structure of the tensor product of vertex op-erator superalagebras. Furthermore, we study the representations of code vertex operator superalgebras resulting from a binary linear code which contains code-words of odd weight. We prove that the code vertex operator superalgebra MD is rational, and Miyamoto's construction of induced modules is generalized to code vertex operator superalgebra. We show that there exists only one set of mutually orthogonal seven conformal vectors with central charge 1/2 in the Hamming code vertex operator superalgebra MH7, and we classify all irreducible MH7-modules. As in [29] and [31], our main tool is the theory of induced modules developed by Dong and Lin in [33] and fusion rules of the rational vertex operator algebra L(1/2,0) with central charge 1/2 (see [26]).
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