Quantum R-matrix and Intertwiners for the Kashiwara Algebra

In a recent work [FR], Frenkel and Reshetikhin developed the theory of q-vertex operators. They showed that n-point correlation functions associated to q-vertex operators satisfy a qdifference equation called q-deformed Kniznik-Zamolodchikov equation. In the derivation of this equation, a crucial point is that the quantum affine algebra is a quasi-triangular Hopf algebra. By using several properties of the quasi-triangular Hopf algebra and the representation theory of the quantum affine algebra, the equation is described in terms of quantum R-matrices.([FR],[IIJMNT]). In [K1], Kashiwara introduced the algebra B q (g), which is generated by 2×rank g symbols with the Serre relations and the q-deformed bosonic relations (See Sect.1,(1.5)) in order to study the crystal base of U, where U is a maximal nilpotent subalgebra of the quantum algebra Uq(g) associated to a symmetrizable Kac-Moody Lie algebra g. (In [K1], B q (g) is denoted by Bq(g)). We shall call this algebra the Kashiwara algebra. He showed that U has a B q (g)-module structure and it is irreducible. He also showed that B q (g) has a similar structure to the Hopf algebra: there is an algebra homomorphism B q (g) → Uq(g) ⊗ B ∨ q (g). Thus if M is a Uq(g)-module and N is a B ∨ q (g)-module, then M ⊗N has a B q (g)-module structure via this homomorphism. In the present paper, we shall first introduce the algebras Bq(g), Bq(g), Uq(g) associated to a symmetrizable Kac-Moody Lie algebra g and algebra morphisms for such algebras. The algebra Bq is obtained by adding the Cartan part to B ∨ q and the algebra Bq is an algebra anti-isomorphic to Bq, where we also call these the Kashiwara algebras. The algebra Uq is an ordinary quantum algebra. The Kashiwara algebra has no Hopf algebra structure, but these algebras admit a certain algebra structure similar to the Hopf algebra. In fact, there are the following algebra homomorphisms, Uq → Uq ⊗ Uq, Bq → Bq ⊗ Uq, Bq → Uq ⊗ Bq, Uq → Bq ⊗ Bq, an antipode S : Uq → Uq and an anti-isomorphism φ : Bq → Bq. By using these, we can consider tensor products and dual modules of Bq-modules, Bq-modules and Uq-modules. (See Sect.1 and Sect.2.) In Sect.2, we discuss properties of the category of highest weight Bq-modules. In Sect.3, we recall the Killing form of Uq due to [R],[T] and give a certain relationship between the algebra B q and the Killing form. We also introduce a bilinear pairing 〈 | 〉 for highest weight Bq-module H(λ), which is an analogue of an ordinary vacuum expectation value. In Sect.4, we restrict ourselves to an affine case and consider the following type of