Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups
Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups
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发表时间：
1997-03
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通讯作者：
V. Tarasov;A. Varchenko
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作者：
V. Tarasov;A. Varchenko
与量子群\(U_q(sl_2)\)相关的三角量子化克尼兹尼克 - 扎莫罗德奇科夫方程（qKZ方程）是一个取值于\(U_q(sl_2)\) Verma模张量积的线性差分方程组。我们用多维\(q\)-超几何函数来求解该方程，并在椭圆量子群\(E_{\rho,\gamma}(sl_2)\)上相应的赋值Verma模的张量积与解空间之间定义了一个自然同构，其中参数\(\rho\)和\(\gamma\)通过\(p = e^{2\pi i\rho}\)以及\(q = e^{-2\pi i\gamma}\)与量子群\(U_q(sl_2)\)的参数\(q\)以及qKZ方程的步长\(p\)相关联。我们构造了与合适的渐近区域相关的渐近解，并根据动态椭圆\(R\)-矩阵计算了渐近解之间的转移函数。这种对转移函数的描述建立了量子环代数\(U_q(\widetilde{gl}_2)\)和椭圆量子群\(E_{\rho,\gamma}(sl_2)\)的表示理论之间的联系，并且类似于关于微分克尼兹尼克 - 扎莫罗德奇科夫方程的单值群的科诺 - 德里菲尔德定理。为了建立这些结果，我们构造了一个离散的高斯 - 曼宁联络，特别是一个合适的离散局部系统、以该局部系统为系数的离散同调与上同调群，并将一个相关的差分方程与qKZ方程等同起来。
The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the quantum group $U_q(sl_2)$ is a system of linear difference equations with values in a tensor product of $U_q(sl_2)$ Verma modules. We solve the equation in terms of multidimensional $q$-hypergeometric functions and define a natural isomorphism between the space of solutions and the tensor product of the corresponding evaluation Verma modules over the elliptic quantum group $E_{\rho,\gamma}(sl_2)$, where parameters $\rho$ and $\gamma$ are related to the parameter $q$ of the quantum group $U_q(sl_2)$ and the step $p$ of the qKZ equation via $p=e^{2\pii\rho}$ and $q=e^{-2\pii\gamma}$. We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the dynamical elliptic R-matrices. This description of the transition functions gives a connection between representation theories of the quantum loop algebra $U_q(\widetilde{gl}_2$ and the elliptic quantum group $E_{\rho,\gamma}(sl_2)$ and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation. In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, discrete homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equation.