Representations Of A Class Of Restricted Forms Of Quantum Groups U_q?sl₂?

The theory of quantum group is a very important research field in mathematics.It arises from theoretical physics and it is closely related to many branches of mathemat-ics.If q is not a root of unity,the(finite-dimensional)representation theory of Uq(sl2)is similar to the case of Lie algebra sl2.If q is a root of unity,the representation theory of Uq(sl2)becomes very complicated.Under the assumption that q is the r-th primitive root of unit,we construct finite dimensional representations of the restricted quantum groups with the relations E2r = 0,Fr = 0,K2r = 1.Firstly,we give some basic properties of the restricted quantum groups Uq(Sl2)and decompose it into the direct sum of their blocks by the degree of generators E,F,K.Then,we construct projective modules Pl of Uq(sl2)by Suter's method of studying quantum groups Liq(sl2).It is more intuitive and clear by using the diagram to describe these modules.If l ? Z/rZ and is odd,we set Pl+r(mod 2r):=Pl.The indecomposable projective representation and their isomorphism classes are given by analysing K-eigenvectors.Then we decompose Uq(sl2)into the direct sum of the principal indecomposable representations.By the property of composition factors,we merge their principle indecomposable left ideas into their blocks.Finally,the decomposition of the tensor product of Uq(sl2)-projective module and simple module is given.