Nakashima Toshiki

We study the algebra B(q)(g) presented by Kashiwara and introduce intertwiners similar to q-vertex operators. We show that a matrix determined by 2-point functions of the intertwiners coincides with a quantum R-matrix (up to a diagonal matrix) and give the commutation relations of the intertwiners. We also introduce an analogue of the universal R-matrix for the Kashiwara algebra.

The explicit form of the crystal graphs for the finite-dimensional representations of the q-analogue of the universal enveloping algebras of type A, B, C, and D is given in terms of semi-standard tableaux and its analogues. (C) 1994 Academic Press, Inc.

We shall give a generalization of the Littlewood-Richardson rule for U(q)(g) associated with the classical Lie algebras by use of crystal base. This rule describes explicitly the decomposition of tensor products of given representations.

We study the higher spin analogs of the six-vertex model on the basis of its symmetry under the quantum affine algebra U(q)(sl2). Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin 1/2, and that the n-particle space has an RSOS type structure rather than a simple tensor product of the one-particle space. This agrees with the picture proposed earlier by Reshetikhin.

The one point functions of the vertex models associated with quantum affine Lie algebras are computed by using the path description of the crystal bases of integral highest weight modules.

We give a basis of the finite dimensional irreducible representation of union-q(X(n)) (X = B, C, D) with highest weight N-LAMBDA-1 (N is-an-element-of Z greater-than-or-equal-to 0,) which we call "symmetric tensor representation". This basis is orthonormal and consists of weight vectors. The action of union-q(X(n)) is given explicitly.