Representation Theory of Elliptic Quantum Groups and the Elliptic q-KZB Equation

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1. The Drinfeld realization of the face type eillptic quantum group B_<q,λ>(g)In the cases g = A^<(1)>_n, A^<(2)>_2, we have realized the L-operator of the face type elliptic quantum group B_<q,λ>(g) in terms of the currents of the elliptic algebra U_<q,p>(g), which is an elliptic deformation of the Drinfeld currents of the affine quantum group U_q(g). Then we have shown the isomorphism U_<q,p>(g) 〓 B_<q,λ>(g) 【cross product】 C{H^^^} as an associative algebra. Here C{H^^^} is a Heisenberg algebra generated mainly by the pair of generators {P_j, Q_j} (j = 1, 【triple bond】, rank g)2. wee field realization of U_<q,p>(g) and algebraic analysis of the solvable lattice modelsThe above isomorphism allows us to construct a free field realization of B_<q,λ>(g). For g = A^<(1)>_n, A^<(2)>_2, we lave constructed the level 1 free field realization of U_<q,p>(g), and obtained a realization of both the finite and infinite dimensional highest weight modules of U_<q,p>(g) and the vertex operator of U_<q,p>(g). These vertex operators are the U_<q,p>(g) counterpart of the B_<q,λ>(g)-intertwining operators. We also have identified the spaces of states in the A^<(1)>_n -type and A^<(2)>_2-type RSOS models with the U_<q,p>(g)-modules. The lance vertex operators in these RSOS models have also been identified with the vertex operators of corresponding U_<q,p>(g)3. Relationship between B_<q,λ>(<sl>^^^^_n) and the deformed W_n-algebraExtending the known relationship between the vertex operators of U_<q,p>(g) for g = A^<(1)>_1, A^<(2)>_2 acid the generating functions of the deformed W(g^^-)-algebras (deformed Virasoro algebra, in these cases), to the higher rank case p = A^<(1)>_<n-1>, we derived the generating functions of the deformed W_n-algebras by calculating the fusion of the level 1 vertex operators of U_<q,p>(<sl>^^^^_n)