Representation theory of Elliptic Quantum Groups and Deformation of W-algebra

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1.Drinfeld realization of B_<q,λ>(g) : For g=A^<(2)>_2, we realized the L-operator of the elliptic quantum group B_<q,λ>(g) in terms of the currents of the elliptic algebra U_<q,p>(g). Then we showed the isomorphism U_<q,p>(g)=^^〜B_<q,λ>(g)【cross product】 C{H^^^} as an associative algebra. Here C{H^^^} denotes some Heisenberg algebra. Futhermore, we constructed the level-1 free field representation of U_<q,p> (A^<(2)>_2), and showed that the vertex operators of U_<q,p>(A^<(2)>_2) provide a realization of the lattice vertex operators of A^<(2)>_2 SOS model in the algebraic analysis formulation.2.B_<q,λ>(sl^^^_n) and the deformed W_n-algebra : Extending the results in the cases g=A^<(1)>_1,A^<(2)>_2 we showed that a fusion of a pair of the level-1 vertex operators of U_<q,p>(sl^^^_n) implies the basic generating function of the deformed W_<n-1>-algebra.3.The Vertex-Face Correspondence and the Elliptic Quantum groups : By using the vertex-face correspondence in solvable lattices models, we give an explicite correspondence of the representations of the two types of the elliptic quantum groups, the vertex type A_<q,p>(sl^^^_2) and the face type B_<q,λ>(sl^^^_2). Based on this, we give an algebraic analysis formulation of the fusion eight-vertex model.4.The Vertex-Face Correspondence and the Elliptic 6j-symbols : We established a new relationship between the vertex-face correspondence intertwining vectors and the elliptic 6j-symbols. We then simplified a proof of the biorthogonal relation, fusion relation and some other relations of the elliptic 6j-symbols. Especially, we derived an equation characterizing the elliptic 6j-symbol, which is similar to the dynamical RLL-relation of the face type elliptic quantum group B_<q,λ>(sl^^^_2).