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

1. 面型椭圆量子群B_<q,λ>(g)的德林菲尔德实现在g = A^<(1)>_n, A^<(2)>_2的情况下，我们用椭圆代数U_<q,p>(g)的电流实现了面型椭圆量子群B_<q,λ>(g)的l算子，它是仿射量子群U_q(g)德林菲尔德电流的椭圆变形。然后我们证明了同构U_<q,p>(g)〓B_<q,λ>(g)【外积】C{H^^^}是一个结合代数。其中C{H^^^}是主要由生成器{P_j， Q_j} (j = 1，【三键】，秩g)2生成的海森堡代数。U_<q,p>(g)的自由场实现和可解晶格模型的代数分析。利用上述同构性，我们可以构造出B_<q,λ>(g)的自由场实现。对于g = A^<(1)>_n, A^<(2)>_2，构造了U_<q,p>(g)的一级自由场实现，得到了U_<q,p>(g)的有限维和无限维最高权模以及U_<q,p>(g)的顶点算子的实现。这些顶点算子是B_<q,λ>(g)-缠结算子的U_<q,p>(g)对应。我们还确定了具有U_<q,p>(g)-模块的A^<(1)>_n型和A^<(2)>_2-型RSOS模型的状态空间。这些RSOS模型中的长矛顶点算子也被识别为对应的U_<q,p>(g)3的顶点算子。B_<q,λ>(<sl>^^^^_n)与变形w_n -代数的关系将已知的当g = A^<(1)>_1, A^<(2)>_2时，U_<q,p>(g)的顶点算子与变形W(g^^-)-代数（变形Virasoro代数，在这种情况下）的生成函数之间的关系推广到更高秩的情况p = A^<(1)>_<n-1>时，通过计算U_<q,p>(<sl>^^^^_n)的一级顶点算子的融合，推导出变形w_n -代数的生成函数

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)