On representations of the elliptic quantum groupE _τ,η(sl ₂)

Abstract

We describe representation theory of the elliptic quantum groupE _τ,η(sl ₂). It turns out that the representation theory is parallel to the representation theory of the YangianY(sl ₂) and the quantum loop group$U_q (\widetilde{sl}_2 )$.

We introduce basic notions of representation theory of the elliptic quantum groupE _τ,η(sl ₂) and construct three families of modules: evaluation modules, cyclic modules, one-dimensional modules. We show that under certain conditions any irreducible highest weight module of finite type is isomorphic to a tensor product of evaluation modules and a one-dimensional module. We describe fusion of finite dimensional evaluation modules. In particular, we show that under certain conditions the tensor product of two evaluation modules becomes reducible and contains an evaluation module, in this case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients. We describe the determinant element of the elliptic quantum group. Representation theory becomes special ifNη=m+lτ, whereN,m,l are integers. We indicate some new features in this case.