Representation And Adjoint Action Of Quantum Algebra U_q(f(K, (?)))

In this paper, let basic field k is complex field C, N denotes the set of non-negative integers, Z⁺denotes the set of positive integers, let q be a parameter with q being not a root of unity. Based on quantum algebras U_q(f(K)) in [5] we construct a algebras U_q(f(K, K)): it is a associative algebras generated by quadruple E, F, K, K satisfying the following relations:It is generalizes the condition K K^-1 = K^-1 K = 1 as KK = K K = J. we give the necessary and sufficient condition of U_q(f(K, K)) have a structure of weak Hopf algebra.Under the definition of weak Ore extension defined by Li Fang, we prove that U_q(f( K, K)) is weak Ore extension of Noetherian ring k[K,K], so U_q(f(K, K)) is a noetherian ring.In this paper we find all finite dimensional integrable highest weight U_q(f(K, K))-module, it is formal as W(n) or V_? and we discuss U_q(f(K, K))-module's Clebsch-Gordan formula:(1)V_a(?)V_b(?)⊙_l=0^min(a,b)V_a+b-2l,(2)V_a(?)(?)(b)(?)(a+1)W(b)．(3)W(b)(?)V_a(?)(a+1)W(b)．(4)W(a)(?)W(b)是平凡模．At last we constrast to adjoint action for quantum algebraωsl_q(2), we prove that U_q(f(K, K)) is a left quasi-module algebra over itself and study the structure of the submodules of F (U_q(f(K, K))), which is the locally finite submodule of the quantum algebra U_q(f(K, K)).