量子包络代数与共形代数的若干研究
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摘要
本文的主要结果分为五个部分.
首先,我们探讨量子包络代数在量子空间上的模代数结构和Ur,t的伴随作用.量子包络代数Ur,t是由吴在[83]中引进的.当q不是单位根且所在的域是复数域C时,我们利用类似于文章[35]中的方法,对Ur,t在量子平面上的模代数结构给出了一个完全的分类,并描述了这些表示.另外,我们还完全分类了Uq(sl(3))在量子3-空间上的模代数结构.并且,当k是特征为0的代数闭域,q∈k不是单位根时,我们对Ur,t的伴随作用进行了研究.我们描述了它的局部有限子代数结构,并刻画了Ur,t的所有理想和它的一些本原理想.
其次,我们研究了对应于量子包络代数U1(f(K,H))(见[81])的由李引进的弱Hopf代数(见[63,64]).在第三章中,我们定义了一类新的代数,记为(?)Uqd.当d=((1,1)|(1,1))时,我们把(?)Uqd记为(?)1Uq;当d=((0,0)|(0,0))时,我们把(?)Ud记为(?)2Uq.并且,我们详细地研究了(?)1Uq和(?)2Uq.在某些情况下,我们给出了(?)1Uq和(?)2Uq成为弱Hopf代数的充分必要条件.(?)1Uq和(?)2Uq的PBW基也已给出.并且,当域是复数域C,q∈C不是单位根时,我们刻画了(?)1Uq的表示和中心.
第三,我们给出了一类新的量子包络代数Uq(f(K,J))和一些新的Hopf代数,这些新的Hopf代数是广义Kac-Moody李代数的量子化包络代数借助任一Hopf代数的某种扩张.这种构造推广了一些已有的在量子化包络代数上加一个Hopf代数的扩张,并且提供了一大类新的非交换非余交换的Hopf代数.
第四,我们引进了左对称共形代数的概念来研究顶点代数.顶点代数是用来描述2维共形场论的一个严格的数学定义.通过Bakalov和Kac在[8]中利用李共形代数和左对称代数给出的顶点代数的等价刻画,我们可以发现,在研究顶点代数时,我们需要处理这样一个问题：是否在一类特殊的李代数(形式分布李代数)上存在与它相容的左对称代数.在第六章中,我们对这个问题进行了研究.左对称共形代数和Novikov共形代数的定义可见第二章.我们在第六章中列举了很多有关这些代数的例子.最后,我们利用左对称共形代数给出了一种构造顶点代数的方法.这种构造提供了一大类有限非交换的顶点代数.
最后,我们讨论了共形意义下的左对称双代数.左对称双代数的定义是由白在[5]引进的,它等价于一个parakahler李代数,这类李代数是带有G不变parakahler结构的李群G的李代数.在第七章中,我们介绍了左对称共形余代数和左对称共形双代数的定义.并且,我们给出了李共形代数和左对称共形代数的匹配对(matched pairs)的构造.我们证明了一个有限的作为C[(?)]模是自由的左对称共形双代数等价于一个parakahler李共形代数(见定义7.18).另外,我们也得到了一个共形意义下的S-方程(见[5]),并给出了共形symplectic double的构造.
The main results of this paper are divided into five parts.
Firstly, we study the module algebra structures of quantum enveloping algebras on the quantum space and the adjoint action of quantum algebra Ur,t-Ur,t is a new quantum enveloping algebra introduced by Wu in [83]. Using the method similar to that in [35], a complete classification of Ur,t-module algebra structures on the quantum plane is given and we describe these representations when q is not the root of unity and the ground field is C. Moreover, we present a complete classification of module algebra structures of Uq(sl(3)) on the quantum3-space. In addition, we investigate the adjoint action of Ur,t when k is a fixed algebraically closed field with characteristic zero and q∈k not a root of unity. The structure of its locally finite subalgebra is given. And, we characterize all its ideals and some primitive ideals of Ur,t.
Secondly, we investigate the weak Hopf algebras introduced by Li corresponding to quantum algebras Ug(f(K, H))(see [81]). In Chapter4, we define a new class of algebras denoted by (?)Uqd. When d=((1,1)|(1,1)), denote (?)Uqd by (?)1Uq; When d=((0,0)|(0,0)), denote (?)Uqd by (?)2Uq. And we study (?)1Uq and (?)2Uq in detail. In some cases, necessary and sufficient conditions for (?)1Uq and (?)2Uq to be weak Hopf algebras are given. The PBW bases of (?)1Uq and (?)2Uq are presented. Finally, representations and the center of (?)1Uq are characterized over C with q∈C not a root of unity.
Thirdly, we present a class of extended quantum enveloping algebras Uq(f(K, J)) and some new Hopf algebras, which are certain extensions of quantized enveloping al-gebras of generalized Kac-Moody Lie algebras by some fixed Hopf algebra H. This con-struction generalizes some well-known extensions of quantized enveloping algebras by a Hopf algebra and provides a large of new non-commutative and non-co-commutative Hopf algebras.
Fourthly, we introduce left-symmetric conformal algebra to study vertex algebra. A vertex algebra is an algebraic counterpart of a two-dimensional conformal field the-ory. By an equivalent characterization of vertex algebra using Lie conformal algebra and left-symmetric algebra given by Bakalov and Kac in [8], in studying vertex algebra, we have to deal with such a question:whether there exist compatible left-symmetric algebra structures on a class of special Lie algebras named formal distribution Lie alge-bras. In Chapter6, we study this question. The definitions of left-symmetric conformal algebra and Novikov conformal algebra are introduced in Chapter2. We show many examples of these algebras in Chapter6. As an application, we present a construction of vertex algebra using left-symmetric conformal algebras. It provides a large of new non-commutative finite vertex algebras.
Finally, we study a conformal analog of left-symmetric bialgebras. The notion of left-symmetric bialgebra was introduced by Bai in [5] which is equivalent to a parakahler Lie algebra which is the Lie algebra of a Lie group G with a G-invariant parakahler struc-ture. In Chapter7, the notions of left-symmetric conformal co-algebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and left-symmetric conformal algebras are presented. We show that a finite left-symmetric conformal bialgebra which is free as a C [(?)]-module is equivalent to a parakahler Lie con-formal algebra (see Definition7.18). We also obtain a conformal analog of the S-equation (see [5]), and give a construction of the conformal symplectic double.
首先,我们探讨量子包络代数在量子空间上的模代数结构和Ur,t的伴随作用.量子包络代数Ur,t是由吴在[83]中引进的.当q不是单位根且所在的域是复数域C时,我们利用类似于文章[35]中的方法,对Ur,t在量子平面上的模代数结构给出了一个完全的分类,并描述了这些表示.另外,我们还完全分类了Uq(sl(3))在量子3-空间上的模代数结构.并且,当k是特征为0的代数闭域,q∈k不是单位根时,我们对Ur,t的伴随作用进行了研究.我们描述了它的局部有限子代数结构,并刻画了Ur,t的所有理想和它的一些本原理想.
其次,我们研究了对应于量子包络代数U1(f(K,H))(见[81])的由李引进的弱Hopf代数(见[63,64]).在第三章中,我们定义了一类新的代数,记为(?)Uqd.当d=((1,1)|(1,1))时,我们把(?)Uqd记为(?)1Uq;当d=((0,0)|(0,0))时,我们把(?)Ud记为(?)2Uq.并且,我们详细地研究了(?)1Uq和(?)2Uq.在某些情况下,我们给出了(?)1Uq和(?)2Uq成为弱Hopf代数的充分必要条件.(?)1Uq和(?)2Uq的PBW基也已给出.并且,当域是复数域C,q∈C不是单位根时,我们刻画了(?)1Uq的表示和中心.
第三,我们给出了一类新的量子包络代数Uq(f(K,J))和一些新的Hopf代数,这些新的Hopf代数是广义Kac-Moody李代数的量子化包络代数借助任一Hopf代数的某种扩张.这种构造推广了一些已有的在量子化包络代数上加一个Hopf代数的扩张,并且提供了一大类新的非交换非余交换的Hopf代数.
第四,我们引进了左对称共形代数的概念来研究顶点代数.顶点代数是用来描述2维共形场论的一个严格的数学定义.通过Bakalov和Kac在[8]中利用李共形代数和左对称代数给出的顶点代数的等价刻画,我们可以发现,在研究顶点代数时,我们需要处理这样一个问题：是否在一类特殊的李代数(形式分布李代数)上存在与它相容的左对称代数.在第六章中,我们对这个问题进行了研究.左对称共形代数和Novikov共形代数的定义可见第二章.我们在第六章中列举了很多有关这些代数的例子.最后,我们利用左对称共形代数给出了一种构造顶点代数的方法.这种构造提供了一大类有限非交换的顶点代数.
最后,我们讨论了共形意义下的左对称双代数.左对称双代数的定义是由白在[5]引进的,它等价于一个parakahler李代数,这类李代数是带有G不变parakahler结构的李群G的李代数.在第七章中,我们介绍了左对称共形余代数和左对称共形双代数的定义.并且,我们给出了李共形代数和左对称共形代数的匹配对(matched pairs)的构造.我们证明了一个有限的作为C[(?)]模是自由的左对称共形双代数等价于一个parakahler李共形代数(见定义7.18).另外,我们也得到了一个共形意义下的S-方程(见[5]),并给出了共形symplectic double的构造.
The main results of this paper are divided into five parts.
Firstly, we study the module algebra structures of quantum enveloping algebras on the quantum space and the adjoint action of quantum algebra Ur,t-Ur,t is a new quantum enveloping algebra introduced by Wu in [83]. Using the method similar to that in [35], a complete classification of Ur,t-module algebra structures on the quantum plane is given and we describe these representations when q is not the root of unity and the ground field is C. Moreover, we present a complete classification of module algebra structures of Uq(sl(3)) on the quantum3-space. In addition, we investigate the adjoint action of Ur,t when k is a fixed algebraically closed field with characteristic zero and q∈k not a root of unity. The structure of its locally finite subalgebra is given. And, we characterize all its ideals and some primitive ideals of Ur,t.
Secondly, we investigate the weak Hopf algebras introduced by Li corresponding to quantum algebras Ug(f(K, H))(see [81]). In Chapter4, we define a new class of algebras denoted by (?)Uqd. When d=((1,1)|(1,1)), denote (?)Uqd by (?)1Uq; When d=((0,0)|(0,0)), denote (?)Uqd by (?)2Uq. And we study (?)1Uq and (?)2Uq in detail. In some cases, necessary and sufficient conditions for (?)1Uq and (?)2Uq to be weak Hopf algebras are given. The PBW bases of (?)1Uq and (?)2Uq are presented. Finally, representations and the center of (?)1Uq are characterized over C with q∈C not a root of unity.
Thirdly, we present a class of extended quantum enveloping algebras Uq(f(K, J)) and some new Hopf algebras, which are certain extensions of quantized enveloping al-gebras of generalized Kac-Moody Lie algebras by some fixed Hopf algebra H. This con-struction generalizes some well-known extensions of quantized enveloping algebras by a Hopf algebra and provides a large of new non-commutative and non-co-commutative Hopf algebras.
Fourthly, we introduce left-symmetric conformal algebra to study vertex algebra. A vertex algebra is an algebraic counterpart of a two-dimensional conformal field the-ory. By an equivalent characterization of vertex algebra using Lie conformal algebra and left-symmetric algebra given by Bakalov and Kac in [8], in studying vertex algebra, we have to deal with such a question:whether there exist compatible left-symmetric algebra structures on a class of special Lie algebras named formal distribution Lie alge-bras. In Chapter6, we study this question. The definitions of left-symmetric conformal algebra and Novikov conformal algebra are introduced in Chapter2. We show many examples of these algebras in Chapter6. As an application, we present a construction of vertex algebra using left-symmetric conformal algebras. It provides a large of new non-commutative finite vertex algebras.
Finally, we study a conformal analog of left-symmetric bialgebras. The notion of left-symmetric bialgebra was introduced by Bai in [5] which is equivalent to a parakahler Lie algebra which is the Lie algebra of a Lie group G with a G-invariant parakahler struc-ture. In Chapter7, the notions of left-symmetric conformal co-algebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and left-symmetric conformal algebras are presented. We show that a finite left-symmetric conformal bialgebra which is free as a C [(?)]-module is equivalent to a parakahler Lie con-formal algebra (see Definition7.18). We also obtain a conformal analog of the S-equation (see [5]), and give a construction of the conformal symplectic double.
引文
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