The Homomorphism And Vanishing Properties Of Induction Modules Of Quantum Algebras

Let R be the root system of type A_n, p be half of the sum of positive roots and Q be a rational field, v is an indeterminate, U_v is the quantum algebra over Q(v) generated by E_i,F_i,K_i,K_i^-1(i=1,2,...,n) and it carries with Q(v)-Hopf structure.Z represents the integral ring, A=Z[v,v^-1].Let U_A be the A sub-algebra of U_v generated by E_i^(m), F_i^(m),K_i^±1 (i=1,2,...,n), then it is an A-Hopf algebra. Suppose k is a field and q∈k is the given non-zero element. v acting on k is equal to q left multiplied, then k becomes an A—module. Define U_q=U_A(?)_Ak, then U_q is a Hopf algebra over k. In this paper, we consider quantum algebra U_q, when q is not a root of 1, Borel-Weil-Bott theory of U_q representation was proved in [2]. When chk = 0, q is the prime power root of 1and weightλ∈X⁺-ρwhich satisfies <λ+p,α>≤l,(?)α∈R⁺ , at this time Borel-Weil-Bott theory sets up, which was proved in [2]. If A is the smallest weight in X⁺-ρ, for any field k, at this time Borel-Weil-Bott theory was also proved in [1]. In this paper, using alcove geometry of affine Weyl group, we proof for any field k , if l is the order of q² and weightλ∈X⁺-ρsatisfiesfor (?)α∈R⁺,<λ+ρ,α>≤l, the above result sets up. To an U_q(i) module Q_i(Ï„) which plays an important role in introducing the four short exact sequences, we discuss the structure and vanishing properties of induction modules of Q_i(Ï„) and itssubmodules. We prove Q_i (Ï„) has an unique irreducible module; Induction module H_q⁰(Q_i (Ï„)) is the irreducible module with the first weightλ. Let K denote the submodule of U_q^b generated by all the weight vectors of Q_i (Ï„) which are smaller thanλ.We also prove that for all j≥0 , H_q^j(K)=0.