Ja n 19 98 A link between two elliptic quantum groups

We consider the category CB of meromorphic finite-dimensional representations of the quantum elliptic algebra B constructed via Belavin’s R-matrix, and the category CF of meromorphic finite-dimensional representations of Felder’s elliptic quantum group Eτ, 2 (gln). For any fixed c ∈ C, we use a version of the Vertex-IRF correspondence to construct two families of (generically) fully faithful functors Hcx : CB → DB and F c x : CF → DB where DB is a certain category of infinite-dimensional representations of B by difference operators. We use this to construct an equivalence between the abelian subcategory of CB generated by tensor products of vector representations and the abelian subcategory of CF generated by tensor products of vector representations. 1 Categories of meromorphic representations In this section, we recall the definitions of various categories of representations of quantum elliptic algebras. Notations: let us fix τ ∈ C, Im(τ) > 0, γ ∈ R\Q and n ≥ 2. Denote by (vi) n i=1 the canonical basis of C and by (Eij) n i,j=1 the canonical basis of End(C ), i.e Eijvk = δjkvi . Let h = { ∑ i λiEii | ∑ i λi = 0} be the space of diagonal traceless matrices. We have a natural identification h = { ∑ i λiE ∗ ii | ∑ I λi = 0}. In particular, the weight of vi is ωi = E ∗ ii − 1 n ∑ k E ∗ kk. Classical theta functions: the theta function θκ,κ′(t; τ) with characteristics κ, κ ∈ R is defined by the formula