Representation theory of the quantized enveloping algebras and the quantized enveloping superalgebras

量化包络代数和量化包络超代数的表示论

基本信息

批准号：
10640022
负责人：
YAMANE Hiroyuki
金额：
$ 2.05万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2001
项目状态：
已结题

项目摘要

Yamane gave a Serre type theorem for the affine Lie superalgebras G, namely he gave a presentation of G by the Chevalley generators and defining relations satisfied by them. He also gave a similar result for the affine quantised superalgebras U_qG. He alos gave a a presentation of U_qG of type A(M|N)^<(1)> by the Drinfeld generators and defining relations satisfied by them, and defining relations satisfied by them. Dnlike the non-super case, the defining relations are very complicate. However, by comparing the defining relations of G with the ones of U_qG, we can find out the coincidense of the dimensions of the weight sapaces of the Verma modules of G with the ones of U_qG. Let R = C[s^<±1>,t^<±1>] be the two variable Laurent polynomials ring. Let D be the universal central extention of sl(2|2). Then dim D/sl(2|2) = 2., and D(R) = D 【cross product】 R 【symmetry】 Ω_R/dR is the universal central extention of sl(2|2) 【cross product】 R. He gave a presentation of D(R) by the finite Chevalle … More y generators and finite definig relations, and also did the same thing for the D type affine Lie superalgebra D^<(1)> = D 【cross product】 C[t^<±1>] + Cc. It is easy to describe the kernel of the natural map D(R)→ sl(2|2)(R) by using the generators. By the fact, we can also give a presentation of sl(2|2)(R) by the finite Chevalley generators and infinite definig relations.Yamane gave a Serre type theorem for the affine Lie superalgebras G, namely he gave a presentation of G by the Chevalley generators and defining relations satisfied by them. He also gave a similar result for the affine quantised superalgebras U_qG. He alos gave a a presentation of U_qG of type A(M|N)^<(1)> by the Drinfeld generators and defining relations satisfied by them. Unlike the non-super case, the defining relations are very complicate. However, by comparing the defining relations of G with the ones of U_qG, we can find out the coincidence of the dimensions of the weight sapaces of the Verma modules of G with the ones of U_qG. Let R = C[s^<±1>,t^<±1>] be the two variable Laurent polynomial ring. Let D be the universal central extension of sl(2|2). Then dim D|sl(2|2) = 2., and D(R) = D 【cross product】 R 【symmetry】 Ω_R/dR is the universal central extension of sl(2|2) 【cross product】 R. He gave a presentation of D(R) by the finite Chevalley generators and finite defining relations, and also did the same thing for the D type affine Lie superalgebra D^<(1)> = D 【cross product】 C[t^<±1>] + Cc. It is easy to describe the kernel of the natural map D(R) → sl(2|2)(R) by using the generators. By the fact, we can also give a presentation of sl(2|2)(R) by the finite Chevalley generators and infinite defining relations.Nagatomo has developed the representation theory of vertex operator algebras, and has applied it to problems arising from conformal field theory. One of the important results is the classification of simple modules for the charge conjugation orbifold model, which opened a way to study conformal field theories with central charge more than or equal to one. On the other hand he applied the systematic study for correlation functions to a construction of modular forms and quasi-modular forms, which attracts much attention of those who work on the theory of modular forms. Less

Yamane 给出了仿射李超代数 G 的 Serre 型定理，即他给出了由 Chevalley 生成元给出的 G 并定义了它们所满足的关系。他还给出了仿射量子化超代数 U_qG 的类似结果。 U_qG 类型为 A(M|N)^<(1)>，由 Drinfeld 生成器定义，并定义它们满足的关系，并定义它们满足的关系。与非超情况一样，定义关系非常复杂，但是通过比较 G 和 U_qG 的定义关系，我们可以发现 G 的 Verma 模的权空间维数与 U_qG 的定义关系是一致的。设 R = C[s^<±1>,t^<±1>] 为二变量洛朗多项式环。设 D 为 sl(2|2) 的通用中心扩张。则dim D/sl(2|2) = 2.，且D(R) = D 【叉积】 R 【对称性】 Ω_R/dR 是sl(2|2) 【叉积】 R 的通用中心扩张。他通过有限 Chevalle … 更多 y 生成器和有限定义关系介绍了 D(R)，并且对 D 型仿射李超代数 D^<(1)> = 做了同样的事情D [叉积] C[t^<±1>] + Cc 使用生成器很容易描述自然映射 D(R)→ sl(2|2)(R) 的核。，我们还可以用有限 Chevalley 生成元和无限定义关系给出 sl(2|2)(R) 的表示。Yamane 给出了仿射李超代数 G 的 Serre 型定理，即他给出了他还给出了仿射量子化超代数 U_qG 的 G 的表示，并定义了它们所满足的关系。他还给出了 A(M|N)^<(1)> 类型的 U_qG 的表示。与非超情况不同，Drinfeld 生成器及其满足的定义关系非常复杂，但是通过比较 G 和 U_qG 的定义关系，我们可以发现。求 G 的 Verma 模的权重空间维数与 U_qG 的重合度，令 R = C[s^<±1>,t^<±1>] 为二变量洛朗多项式环。 D 是 sl(2|2) 的通用中心扩展，则 dim D|sl(2|2) = 2.，并且 D(R) = D [叉积] R [对称性] Ω_R/dR 是通用中心扩展。 sl(2|2) 【叉积】 R 的中心扩展。他用有限 Chevalley 生成元和有限定义关系给出了 D(R) 的表示，并对 D 型仿射李超代数 D^< (1)> = D 【叉积】 C[t^<±1>] + Cc 使用以下公式很容易描述自然映射 D(R) → sl(2|2)(R) 的核。事实上，我们还可以用有限Chevalley 生成器和无限定义关系给出sl(2|2)(R) 的表示。Nagatomo 发展了顶点算子代数的表示理论，并将其应用于所出现的问题。其重要成果之一是电荷共轭轨道模型的简单分类，为研究中心电荷大于或等于1的共形场论开辟了道路。对相关函数与模形式和拟模形式的构造的系统研究，引起了模形式理论研究人员的广泛关注。