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

1.B_<q,λ>(g)的Drinfeld实现：对于g=A^<(2)>_2，我们用椭圆代数U_<q,p>(g)的电流实现了椭圆量子群B_<q,λ>(g)的l算子。在此基础上，我们证明了同构U_<q,p>(g)=^^ ~ B_<q,λ>(g)【外积】C{H^^}是一个结合代数。这里C{H^^^}表示某种海森堡代数。在此基础上，构造了U_<q,p>(A^<(2)>_2)的一级自由域表示，并证明了U_<q,p>(A^<(2)>_2)的顶点算子提供了A^<(2)>_2 SOS模型的格顶点算子在代数分析公式中的实现。B_<q,λ>(sl^^ _n)和变形的w_n -代数：推广了g=A^<(1)>_1,A^<(2)>_2情况下的结果，证明了U_<q,p>(sl^^ _n)的一对一级顶点算子的融合可以得到变形的W_<n-1>-代数的基本生成函数。顶点-面对应与椭圆量子群：利用可解格模型中的顶点-面对应，给出了顶点类型A_<q,p>(sl^^ _2)和面类型B_<q,λ>(sl^^ _2)两类椭圆量子群表示的显式对应关系。在此基础上，给出了融合八顶点模型的代数解析表达式。顶点-面对应与椭圆6j符号：建立了顶点-面对应交织向量与椭圆6j符号之间的新关系。然后，我们简化了椭圆型6j符号的双正交关系、融合关系和其他一些关系的证明。特别地，我们导出了一个与面型椭圆量子群B_<q,λ>(sl^^^_2)的动态rll关系相似的椭圆6j符号的方程。

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).