Nonweight Representations Of The Quantum Group U_q?sl₂?

Let g be a finite-dimensional semisimple Lie algebra,and q ? F is nonzero and not a root of unity.The quantum group Uq(g)is the quantization of the universal enveloping algebra,of g.The representation theory of quantum groups has important applications in theoretical physics and other mathematical branches.In this paper,we study some nonweight representations of the simplest quantum group Uq(sl2),where sl2 is the 3-dimensional simple lie algebra.More precisely,we construct-ed Uq(sl2)-module structures on the Laurent polynomial space C[K±1]with the action of U(CK±1)being free,then we determined the irreducibility of these modules.We proved that any Uq(sl2)-module which admits a rank-1 free U(CK±1)-module structure is isomor-phic to one of the modules we constructed.At last,we investigated the tensor product of the above defined modules with finite-dimensional modules of Uq(sl2)and obtained a.decomposition formula similar to the well known Clebsch-Gordan formula.