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

Yamane给出了仿射李超代数G的一个Serre型定理，即他给出了用Chvalley生成元表示G，并定义了它们所满足的关系。他对仿射量化超代数U_QG也给出了类似的结果。他还给出了A(M|N)^&lt；(1)&gt；型U_QG的Drinfeld生成元，定义了它们满足的关系，定义了它们满足的关系。与非超级情况一样，定义关系非常复杂。然而，通过比较G和U_QG的定义关系，我们可以发现G的Verma模的权空间的维度与U_QG的权空间的维度是一致的。设R=C[S^&lt；±1&gt；，t^&lt；±1&gt；]是二元洛朗多项式环。设D是sl(2|2)的泛中心扩张。D(R)=D[叉积]R[对称性]Ω_R/dR是sl(2|2)[叉积]R的泛中心扩张。给出了D(R)的有限Chevle…表示对D型仿射李超代数D^&lt；(1)&gt；=D[叉积]C[t^&lt；±1&gt；]CC也做了同样的事情。利用生成元可以很容易地描述自然映射D(R)→sl(2|2)(R)的核。Yamane给出了仿射李超代数G的Serre型定理，即他给出了仿射李超代数G的Chvalley生成元的表示及其满足的关系.他对仿射量化超代数U_QG也给出了类似的结果。他还给出了A(M|N)^&lt；(1)&gt；型U_QG的Drinfeld生成元及其满足的关系的定义。与非超级情况不同，定义关系非常复杂。然而，通过比较G和U_QG的定义关系，我们可以发现G的Verma模的权空间的维度与U_QG的权空间的维度是一致的。设R=C[S；±1&gt；，t^&lt；±1&gt；]是二元洛朗多项式环。设D是sl(2|2)的泛中心扩张。然后D(R)=D[叉积]R[对称性]Ω_R/DR是sl(2|2)[叉积]R的泛中心扩张。他利用有限Chvalley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^&lt；(1)&gt；=D[叉积]C[t^&lt；±1&gt；]CC也做了同样的事情。利用生成元可以很容易地描述自然映射D(R)→sl(2|2)(R)的核。利用有限Chvalley生成元和无限定义关系，给出了sl(2|2)(R)的表示。Nagtom发展了顶点算子代数的表示理论，并将其应用于共形场理论中的问题。其中一个重要结果是对电荷共轭奥比福尔德模型的简单模进行了分类，为研究中心电荷大于或等于1的共形场论开辟了一条新的途径。另一方面，他将相关函数的系统研究应用于模形式和拟模形式的构造，这引起了模形式理论研究人员的极大关注。较少

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