Representations Of Quantum Algebra U_q(f_m(K, H)) And Its (?)-form U_(?)

Quantum group is an important branch of algebra. (?)nd the theory about the quantumgroups has been widely studied in recent years. The quantum group U_q(f(K, H)) introducedby professor Wang Dingguo in 2002 is a natural generalization of the quantum group U_q(sl₂),which is the quantized enveloping algebra of U(sl₂). Based on the quantum group U_q(f(K, H)),this paper mainly studies the related content about the quantum group when In this case, the quantum group is denoted by U_q(f_m(K, H)).In this paper, let k is an algebraically closed field of characteristic 0, k0 is a subfield of k.q is nonzero and is not a root of unity. Let N denote the set of natural numbers and Z denote theset of integers.We firstly introduce the definition of the quantum group U_q(f_m(K, H)) , study its Hopfalgebraic structure and its finite dimensional representations, and also obtain that the categoryof all finite dimensional U_q(f_m(K, H))-weight modules of type 1 is equivalent to those of typeα, whereαis a m-th root of unity. In particular, we get the quantum Clebsch-Gordan formulaabout the finite-dimension U_q(f_m(K, H))-weight modules.Secondly, we construct the (?) = k₀[q, q^-1] form of U_q(f_m(K, H)), which is denoted byU(?), and give its Hopf algebraic structure and its triangular decomposition as vector spaces.Finally, we study the representation theory of the algebra U(?) . We give the relationship be-tween U_q-modules Vα(b, d)(b∈k^*, d∈N) and U(?) -modules V _?^α(b, d) = U(?) , where V^α(b, d)is the finite simple module which is generated by a highest weight vector v of weight (αbq2d, b).Using the theory of the classical limit, we mainly show that V has a U(sl₂)-module structure,where V = V_?^α(b, d)(?) k₀, and then we study the relationship between the finite-dimensionsimple modules of U(?) and those of U(sl₂). At the same time, we also obtain the relationshipbetween their V erma-modules.