Finite Dimensional Representations Of Uq(S12)

The concept of quantum group was first introduced by the former Soviet mathematician Drinfeld in1985. In1986, Drinfeld made a report that the contents of the report about the quantum group at the International Congress of mathematicians, and, in1990, Drinfeld awarded the Fields Prize. Since then, Quantum group has become the hot subject of international mathematics. Quantum group that it from generation to now only have a few tens of years, quantum group is a newly developed subject with a lot of scope for development. Many experts and scholars from home and abroad are greatly focusing on the researching work of the quantum group theroy. In recent years, the research of the theory of quantum group Uq (sl2) has made great progress. In the study of the theory of quantum groups Uq (sl2), It is well know that, when q is not a root of unity, the (finite-dimensional) representation theory of Uq (sl2) is similar to that of Lie algebra sl2, and it is completely determined.when q is a root of unity, the representation theory of Uq(sl2) becomes complex and diffcult. Due to the Finite Dimensional Representations of quantum group Uq (sl2) can be regarded the Representations of quotient algebra Uq(m,n,b) of quantum group Uq(sl2). In order to study a class of Finite Dimensional Representations of quantum group Uq(sl2), we can turn to study Representations of quotient algebra Uq(m,n,b) of quantum group Uq(sl2). So it is very important to study Representations of quotient algebra Uq(m,n,b) of quantum group Uq(sl2) Although in recent years the research of quantum group theory has made great progress, but as a new subject of quantum groups still exist a lot of space for research, and in this area there are a lot of problems have not been solved. One of the most basic problem is to determine all finite dimensional representation of quantum group Uq(sl2).In order to study a class of Finite Dimensional Representations of quantum group Uq(sl2), we study Uq(sl2) with relations Kr=1, Emr=b,Fnr=0.So we turn to quotient algebra Uq(m,n,b) of quantum group Uq(sl2) with relations Kr=1, Emr=b,Fnr=0is deal with in this paper, where q is a root of unity. Fristly, we define Uq(m,n,b) and give its basic properties. The quotient algebra Uq(m,n,b) of quantum group Uq(sl2) is the associative C-algebra with1generated by E, F,K,K-1subject to the relations:(R1),(R2),(R3),(R4),(R5),(R6)in this paper. Secondly, we study finite dimensional modules M over quantum group Uq(sl2) satisfying relations Krz=z, Emrz=bz for all z∈M and Fnr M=0. These modules can be regarded as modules over the quotient algebra Uq(m,n,b) of quantum group Uq(sl2) with relations Kr=1, Emr=b and Fnr=0. We construct projective modules of Uq(m,n,b), and decompose Uq(m,n,b) into a direct sum of principal indecomposable modules. We begin with the left ideal Ua1(1) of Uq(m,n,b). The construction of projective modules for0≤l<r-1and l=r-1are different, We discuss these two case separately. Finally, we combine PIMs into blocks and study their structure. Therefore, the problem of representations of Uq(m,n,b) is reduced to those of algebra. Therefore, the problem of finite dimensional representations of Uq(m,n,b) is reduced to those of Al, A2C[x]/(xn) and C[x,y]/(xn,ym-1).