Differential Equations Compatible with KZ Equations

By Giovanni Felder, Y. Markov, V. Tarasov, A. Varchenko. 2000

143 citations recorded by the source corpus.

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 g. These are equations on a function of n complex variables zi taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the ‘dual’ variable in the Cartan subalgebra of 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c4858783c9229e9ec43efcb43ad76ffe0fb97663

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. P. Boalch
2. Valerio Toledano Laredo
3. V. Tarasov
4. A. Varchenko
5. F. Alberto Grünbaum
6. M. Yakimov
7. John Millson
8. A. J. Durán

In Geometry and Topology

1. P. Boalch
2. V. Tarasov
3. Valerio Toledano Laredo
4. A. Varchenko
5. F. Alberto Grünbaum
6. M. Yakimov
7. John Millson
8. Daisuke Yamakawa

In Statistical and Nonlinear Physics

1. Valerio Toledano Laredo
2. V. Tarasov
3. A. Varchenko
4. P. Boalch
5. F. Alberto Grünbaum
6. M. Yakimov
7. Alexander Pushnitski
8. David Damanik

Citations in the corpus

• Casimir Operators and Monodromy Representations of Generalised Braid Groups (2003, background): “Unlike ∇κ however, the connection introduced in [ FMTV ] is not W -equivariant and therefore only defines representations of the pure braid group Pg instead of the full braid group Bg CASIMIR OPERATORS AND GENERALISED BRAID GROUPS 219”
• A Kohno-Drinfeld theorem for quantum Weyl groups (2002, methodology): “The aim of this paper is to show that these quantum Weyl group representations describe the monodromy of the flat connection introduced in [MTL] and, independently, in [ FMTV ].”
• Discrete bispectral Darboux transformations from Jacobi operators (2000, background): “These include automorphisms and ideal structure of the Weyl algebra in one variable [5, 3], representations of the W1+∞ algebra [2], Calogero–Moser system [25], Huygens’ principle [4], traces of intertwiners for representations of (quantized) simple Lie algebras [8, 9] (the last two in the multivari”
• Irregular connections and Kac-Moody root systems (2008, methodology): “(Recall that Reshetikhin [29] explained how to obtain the KZ equations from the Fuchsian isomonodromy equations and [8] Prop.4.4 explained how to obtain the extension [ 17 , 32] of KZ responsible for the quantum Weyl group actions, from the simplest irregular isomonodromy equations.)”
• Duality for Knizhnik–Zamolodchikov and Dynamical Equations (2001, methodology): “The statement follows from the same result for the slk case in [ FMTV ].”