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

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640022/
• 关键词: Superalgebras; Quantum groups; Toroidal superalgebras; Vertex operator algebras; Representaion theory; Number theory; スーパーリー代数; ホップ代数; 表現論; 数理物理学; 偏微分方程式; 量子包絡超代数; 岩堀ヘッケ代数

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给出了一个塞尔型定理的仿射李超代数G，即他提出了G的Chevalley发电机和定义的关系满足他们。他也给出了类似的结果仿射量子超代数U_qG。并给出了A（M）型U_qG的一个表示|N）^<（1）>，并定义了它们所满足的关系。与非超情形不同，定义关系非常复杂。然而，通过比较G与U_qG的定义关系，我们可以发现G的Verma模的权空间的维数与U_qG的权空间的维数一致。设R = C[s^<±1>，t^<±1>]是二元Laurent多项式环.设D是sl（2）的泛中心扩张|2）的情况。然后调暗D/sl（2| 2）= 2.，且D（R）= D [叉积] R [对称性] Ω_R/dR是sl（2）的泛中心扩张|2）[叉积] R.他介绍了D（R）的有限Chevalle ...更多信息 y生成元和有限定义关系，并对D型仿射李超代数D^<（1）> = D [叉积] C[t^<±1>] + Cc也做了同样的工作.自然映射D（R）→ sl（2）的核容易刻画|（2）（R）使用发电机。事实上，我们也可以给出sl（2| Yamane给出了仿射李超代数G的一个Serre型定理，即用Chevalley生成元表示G，并定义了它们所满足的关系。他也给出了类似的结果仿射量子超代数U_qG。并给出了A（M）型U_qG的一个表示|N）^<（1）>，并定义了它们所满足的关系。与非超情形不同，定义关系非常复杂。然而，通过比较G与U_qG的定义关系，我们可以发现G的Verma模的权空间的维数与U_qG的权空间的维数是一致的。设R = C[s^<±1>，t^<±1>]是二元Laurent多项式环.设D是sl（2）的泛中心扩张|2）的情况。然后调暗D| sl（2| 2）= 2.，D（R）= D [叉积] R [对称性] Ω_R/dR是sl（2）的泛中心扩张|2）[叉积] R.他利用有限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也做了同样的工作.自然映射D（R）→ sl（2）的核容易刻画|（2）（R）使用发电机。事实上，我们也可以给出sl（2| Nagatomo发展了顶点算子代数的表示理论，并将其应用于共形场论中的问题。其中一个重要的结果是电荷共轭轨道模型的简单模的分类，它为研究中心电荷大于或等于1的共形场论开辟了一条道路。另一方面，他应用系统的研究相关函数的建设模形式和准模形式，这吸引了许多人的注意，谁的工作理论的模形式。少

项目成果

H. Yamane: "On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras"Publ. RIMS Kyoto UNIV.. 35 (3). 321-390 (1999)

H. Yamane：“关于仿射李超代数和仿射量子化泛包络超代数的关系的定义”Publ。

村上 斉: "The colored Jones polynomials and the simplicial volume of a knoe"Acta Mathematica. 186. 85-104 (2001)

Hitoshi Murakami：“彩色琼斯多项式和节点的单纯体积”Acta Mathematica 186. 85-104 (2001)。

Takao Watanabe: "On an analog of Hermite's constant"J.Lie Theory. (to appear).

Takao Watanabe：“关于厄米常数的类比”J.Lie 理论。

Chongying Dong: "Classification of irreclucible modules for the vertex operator algebra M(1)^+"J.Algera. 216・1. 384-404 (1999)

董崇英：“顶点算子代数 M(1)^+ 的不可分解模的分类”J.Algera 216・1 (1999)。

大槻 知忠: "量子不変量:3次元 トポロジーと数理物理の遭遇"日本評論社. 170 (1999)

大月智忠：“量子不变量：三维拓扑与数学物理的相遇”Nippon Hyoronsha 170 (1999)。