| In this paper, let U = Uq(f(K)) be a generalization of the quantized enveloping algebra Uq(sl(2)), and F(U) be the locally finite subalgebra of U under the adjoint action. Assume I is a ideal of F(U), then I is called a stable ideal, if it is a U- submodule of F(U) under the adjoint action.We show that every non-zero stable ideal of F(U) can be generated by a sum of some highest weight vectors. Denote by I the non-zero stable ideal of F(U), then I= 1(Cq)En1K-n1s+ g2(Cq)En2K-n2s+…+ gt(Cq)EntK-nts) for some polynomialsgi(Cq)∈k[Cq] and 1≤i≤t.For the case f(K) =Km-K-m/q-q-1for m∈N and q is not a root of unity. Using the well-known representation theory, and the structures of locally finite subalgebra and the properties of annihilator polynomials of simple modules, we mainly study the non-zero ideals of U and prove that every non-zero ideal of U can be generated by two highest weight vectors under the adjoint action, and by a sum of the two highest weight vectors. let I be a non-zero ideal of U, then there exist an integer n≥0 and f(Cq), g(Cq)∈k[Cq] with g(Cq) satisfying g(Cq) |fn0(Cq) andφn(Cq) | g(Cq) such that I= (EnK-nsf(Cq), f(Cq)g(Cq)) = (EnK-nsf(Cq) + f(Cq)g(Cq)). On the base of this, the weight property make it possible to give a complete list of all prime (primitive, maximal) ideals of U according to their generators. Moreover, it turns out that every non-zero ideal of U can be uniquely written as a product of primes. |
