Differential Equations Compatible with KZ Equations
Abstract
We define a system of "dynamical" differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra $\mathbf{g}$. These are equations on a function of $n$ complex variables $z_i$ taking values in the tensor product of $n$ finite dimensional $\mathbf{g}$modules. The KZ equations depend on the "dual" variable in the Cartan subalgebra of $\mathbf{g}$. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2000
 arXiv:
 arXiv:math/0001184
 Bibcode:
 2000math......1184F
 Keywords:

 Quantum Algebra;
 35Q40;
 17B10;
 17B56;
 81R10;
 33C80
 EPrint:
 34 pages, LaTeX