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

1.B_&lt；q，λ&gt；(G)的Drinfeld实现：对于g=A^&lt；(2)&gt；_2，我们利用椭圆代数U_&lt；q，p&gt；(G)的流实现了椭圆量子群B_&lt；q，λ&gt；(G)的L算符。然后我们证明了同构U&lt；q，p&gt；(G)=^~B&lt；q，λ&gt；(G)[叉积]C{H^}是结合代数。这里C{H^^}表示一些Heisenberg代数。进一步，我们构造了U_&lt；q，p&gt；(A^&lt；(2)&gt；_2)的第一级自由场表示，并证明了U_&lt；q，p&gt；(A^&lt；(2)&gt；_2)的顶点算子在代数分析公式中提供了A^&lt；(2)&gt；_2SOS模型的格点算子的实现。2.B_&lt；Q，λ&gt；(SL^^n)与变形W_n-代数：推广了g=A^&lt；(1)&gt；_1，A^&lt；(2)&gt；_2情形下的结果，证明了U_&lt；q，p&gt；(sl^^n)的一对1级顶点算子的融合蕴含着变形W_&lt；n-1&gt；3.点-面对应与椭圆量子群：利用可解格模型中的点-面对应，我们给出了两类椭圆量子群，即顶点A_&lt；q，p&gt；(sl^^_2)和面型B_&lt；q，λ&gt；(sl^^2)的表示的显式对应。在此基础上，给出了融合八顶点模型的代数分析公式。4.点-面对应与椭圆6J-符号：建立了点-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，我们导出了一个表征椭圆6J符号的方程，它类似于面型椭圆量子群B_&lt；q，λ&gt；(sl^2)的动力学RLL关系。

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).