Classification Of Quantum Group A_n-Module Algebra Structures On Quantum Polynomial Algebras

The study of quantum polynomial algebras was initiated by McConnel and Pettit [1] as a multiplicative analog of the Weyl algebra. It is well known that quantum polynomial algebras are a very important class of noncommutative al-gebra and the basis for study of noncommutative algebraic geometry. During the latest30years, people pay more attention to the study of quantum groups. The theory of quantum groups has deeply linked with many areas of mathematics and directly promoted the development of physics. Due to the close relationships between quantum polynomial algebras and quantum groups, and a vital tool for the study of quantum groups, the research of quantum polynomial algebras not only greatly promote the development of the theory of quantum groups them-selves, but also add new research content for algebraic representation theory. Quantum polynomial algebras also have remarkable applications in the theory of theoretical physics, low-dimensional topology and other fields. Especially in recent years, quantum polynomial algebras are paid more and more attention by mathematical workers. There are many fruitful results. For example, Artamonov et al. in the literatures [2,3,4,5,6,7,8,9,10] have done a series of research work in quantum polynomial algebras. They have studied many properties of quantum polynomial algebras and some related theory of quantum polynomial algebras as representations of quantum groups. Quantum plane as a simplest quantum polynomial algebra, it is used more widely because of its simple struc-ture. Kassel in [11] gave a kind of module algebra structures of quantum group Uq(sl2) acting on the quantum plane. Duplij and Sinel’shchikov [12] studied the classification of quantum group Uq(sl2)-module algebra structures on the quan-tum plane. The structures that exist on the quantum plane are widely used as a background to produce associated structures for more sophisticated quantum algebras. Furthermore, there are close relationships between the classification of module algebras and the category of modules in tensor category of quantum groups. Etingof and Ostrik et al. did much research work, see [13,14,15] and so on. Recently, Zhang et al.[16] discussed the non toric Uq(sl2)-module algebra structures on Fq[x±1,y]. Duplij and Sinel’shchikov in their paper [12] proposed the following question: It would be interesting to apply this approach to produce a similar classification for Uq(sln+1)-module algebra structures, and even in the case of more general quantum algebras. The most complicated automorphism of quantum polyno-mial algebras is the one containing three variables. Inspired by research work of Duplij et al., this doctoral dissertation investigates the classification of quan-tum group A1-module algebra structures on quantum polynomial algebras with three variables and quantum group An-modulc algebra structures on the quantum plane. Throughout the doctoral dissertation, suppose Z is an integral domain, C is a complex field, the parameter q E C, and q is an indeterminate over Z. In this dissertation, we first investigate the classification of Uq(sl2)-module alge-bra structures on Cq[x,y,z]. Then, suppose n>1, we discuss the classification of Uq(sln+1)-module algebra structures on the quantum plane. The dissertation is divided into four chapters. In the following, we will introduce the specific arrangements of the thesis.In Chapter One, we review some basic concepts and results. We briefly recall some notions about Hopf algebras and their modules, module algebras, quantum group Uq(sln+1) and its related representation theory, and the definition of quantum polynomial algebras and so on.In Chapter Two. when the parameter t in the automorphism of Cq[x, y. z] is equal to0, we investigate the classification of Uq(sl2)-module algebra structures on the quantum polynomial algebra Cq[x,y,z]. Similar to the discussions of Duplij and Sinel’shchikov in [12], we introduce a full action matrix M, an action K-matrix, an action EF-matrix and the i-th homogeneous component symbol matrix and etc. One knows that every monomial of Cq[x, y, z] is an eigenvector for K, and one can calculate the associated eigenvalue. Therefore, we obtain some relations between full action matrixes. Moreover, according to the0-th homogenous component of the full action matrix M and some relations, one can get7cases of (Mef)0,6of which determine the specific values of α, β and γ. In a similar way, we also consider the1-th homogeneous component of M, and receive15cases of (Mef)1. There are14cases of (Mef)1fix the precise values of α,β and γ. Then we introduce symbolic matrix series. Because each such matrix determines a pair of specific weight constants α,β and γ, and we exclude all the empty series. Finally, we turn to discuss the remaining11non-empty series one by one. Consequently, we obtain the classification Uq(sl2)-module algebra structures on Cq[x, y, z] under the assumption t=0.In Chapter Three, when the parameter t in the automorphism of Cq[x, y, z] is not equal to0, we study the classification of Uq(sl2)-module algebra structures on Cq[x,y,z]. The method is similar to the third chapter. For the sake of brevity, we also introduce a full action matrix M, an action K-matrix, an action EF-matrix and the i-th homogeneous symbol matrix and so on. Obviously, we know that any monomial xmzp∈Cq[x,y,z] is an eigenvector of K and we can calculate the associated eigenvalue. So we get the relationships between symbol matrixes. Then, similar to chapter two, we discuss all the possibilities of (Me/)o and (Mef)I. According to the values of α,β and γ, one has31symbolic matrix series,16of which are empty series. We prove the left15symbolic matrix series one by one, and we also get14empty series as well. Finally, when t≠0, we gain the classification of Uq(sl2)-module algebra structures on Cq[x, y, z].In Chapter Four, suppose n>1, we discuss quantum group Uq(sln+1)-module algebra structures on the quantum plane. We first recall the classi-fication of Uq(sl2)-module algebra structures on the quantum plane in Duplij and Sinel’ shchikov [12]. They have obtained6classes of Uq(sl2)-vaodule alge-bra structures. Then, according to the relationships between quantum groups Uq(sln+1) and Uq(sl2), we verify those6types of module algebras one by one. Finally, there is only one type module algebra structures satisfy all the relation-ships of the quantum group Uq(sln+1). Therefore, we obtain the quantum group Uq(sln+1)-module algebra structures on the quantum plane.