Reasearch On The Dominant Weights With The Borel-Weil-Bott Property

Let A=Z[v, v-1], U is a quantum algebra on A. It's an A-Hopf algebra defined by generators and relations. Let k is a field, q is a nonzero element in k. A→k(U(?)q) is A algebra homomorphism. Let Uq=U(?)Ak, then Uq is a quantum algebra on k. It has k-Hopf structure. The induced representation of the quantum algebras and the vanishing property of the high-level homological module on it are the important contents about the reasearch of the representation of quantum algebras. In order to make a deeper reasearch on the induced representation of the quantum algebras and the vanishing property, this paper introduces a new concept—dominant weight with the Borel-Weil-Bott property (B-W-B property for short) on the minimal parabolic subalgebras. We proves that ifλhas the B-W-B property on the minimal parabolie subalgebras, thenλsatisfies the Borel-Weil-Bott theorem on Uq. Based on the preceding theorem this paper also proves that Hq0(λ)is a irreducible Uq module ifλhas the B-W-B property. This paper gives the character of the dominant weight with the Borel-Weil-Bott property and describes the states of their distrubutions. This paper also studies the non-regular dominant weight with the Borel-Weil-Bott property for the quantum algebras of type A1,A2, and provides some dominant weights of this type.