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

椭圆量子群的表示论和椭圆q-KZB方程

• 批准号: 14540028
• 负责人: EGUCHI Masaaki
• 金额: $ 1.54万
• 依托单位: HIROSHIMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540028/
• 关键词: Quantum Group; Elliptic Function; Conformal Field Theory; Lie algebra; Hopf algebra; Solvable Lattice Model; 表現; 頂点作用素; Lie環; アフィンLie環

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)

1. 面型椭圆量子群 B_<q,λ>(g)的 Drinfeld 实现在 g = A^<(1)>_n, A^<(2)>_2 的情况下，我们实现了 L-面型椭圆量子群 B_<q,λ>(g) 的算子用椭圆代数 U_<q,p>(g) 的电流表示，它是椭圆变形仿射量子群 U_q(g) 的德林菲尔德电流的同构 U_<q,p>(g) 〓 B_<q,λ>(g) 【叉积】 C{H^^^}作为结合代数。这里 C{H^^^} 是主要由生成元对 {P_j, Q_j} (j = 1, [triple] 生成的海森堡代数。键】，秩 g)2. U_<q,p>(g) 的场实现和可解晶格模型的代数分析上述同构允许我们构造 B_<q,λ>(g) 的自由场实现。对于g = A^<(1)>_n, A^<(2)>_2，我们构造了U_<q,p>(g)的1级自由场实现，并得到了两者的实现U_<q,p>(g) 的有限维和无限维最高权重模块以及 U_<q,p>(g) 的顶点算子 这些顶点算子是 U_<q,p>(g) 的对应项。我们还使用 A^<(1)>_n 类型和 A^<(2)>_2 类型 RSOS 模型识别了 B_<q,λ>(g) 交织算子的状态空间。 U_<q,p>(g)-模块也已确定了这些 RSOS 模型中的矛顶点算子与相应 U_<q,p>(g)3 之间的关系。 <sl>^^^^_n) 和变形的 W_n-代数扩展了 U_<q,p>(g) 的顶点算子之间的已知关系，对于 g = A^<(1)>_1， A^<(2)>_2 将变形 W(g^^-)-代数（在这些情况下为变形 Virasoro 代数）的生成函数酸化为更高阶的情况 p = A^<(1)>_< n-1>，我们通过计算 1 级顶点算子的融合导出了变形 W_n 代数的生成函数U_<q,p>(<sl>^^^^_n)

项目成果

T.Kojima et al.: "The elliptic algebra U_{q, p}(\hat{sl}_N)and the Drinfeld realization of the elliptic quantum group B_{q,\lambda}(\hat{sl}_N)"Communications in Mathematical Physics. 239. 405-447 (2003)

T.Kojima 等人：“椭圆代数 U_{q, p}(hat{sl}_N) 和椭圆量子群 B_{q,lambda}(hat{sl}_N) 的 Drinfeld 实现”

T.Kojima, H.Konno: "The elliptic algebra U_{q,p}(\widehat{{sl}}_N)$ and the Drinfeld realization of the elliptic quantum group ${\cal B}_{q,\lambda}(\widehat{{sl}}_N)"Comm.Math.Phys.. 239. 405-447 (2003)

T.Kojima、H.Konno：“椭圆代数 U_{q,p}(widehat{{sl}}_N)$ 和椭圆量子群 ${cal B}_{q,lambda 的 Drinfeld 实现

K.Takasaki et al.: "An integrable system on the moduli space of rational functions and its variants"J.Geom.Phys.. 47. 1-20 (2003)

K.Takasaki 等：“有理函数模空间上的可积系统及其变体”J.Geom.Phys.. 47. 1-20 (2003)

B.Feigin et al.: "Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials."Int.Math.Res.Not.. 18. 1015-1034 (2003)

B.Feigin 等人：“对称多项式在移动对角线上消失和麦克唐纳多项式。”Int.Math.Res.Not.. 18. 1015-1034 (2003)

M.Ebata et al.: "The Cowling-Price theorem for semisimple Lie groups""Hiroshima Math.J.. 32. 337-349 (2002)

M.Ebata 等：“半单李群的 Cowling-Price 定理””Hiroshima Math.J.. 32. 337-349 (2002)