Differential Equations Compatible with KZ Equations
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 z i 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.
- Babujian, H. and Kitaev, A.: Generalized Knizhnik-Zamolodchikov equations and isomonodromy quantization of the equations integrable via the inverse scattering transform: Maxwell-Bloch system with pumping, J. Math. Phys. 39 (1988), 2499–2506.
- Chalykh, O. A., Feigin, M. V. and Veselov, A. P.: New integrable generalizations of Calogero-Moser quantum problem, J. Math. Phys. 39 (1998), 695–703.
- Chalykh, O. A. and Veselov, A. P.: Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 126 (1990), 597–611.
- Drinfeld, V.: Quantum groups, in: Proc. ICM (Berkeley, 1986), Vol. 1, Amer. Math. Soc., Providence, RI, 1987, pp. 798–820.
- Duistermaat, J. J. and Grünbaum, F. A.: Differential operators in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240.
- Douai, A. and Terao, H.: The determinant of a hypergeometric period matrix, Invent. Math. 128 (1997), 417–436.
- Harnad, J. and Kasman A. (eds.): The Bispectral Problem (Montréal, 1997), CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, RI, 1998.
- Markov, Y., Tarasov, V. and Varchenko, A.: The determinant of a hypergeometric period matrix, Houston J. Math. 24(2) (1998), 197–219.
- Orlik, P. and Solomon, L.: Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189.
- Schechtman, V. and Varchenko, A.: Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139–194.
- Varchenko, A.: Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum groups, Adv. Ser. Math. Phys. 21, World Scientific, Singapore, 1995.
- Wilson, G.: Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177–204.
- Ziegler, G.: Matroid shellability, β-systems, and affine arrangements, J. Algebraic Combin. 1 (1992), 283–300.
- Differential Equations Compatible with KZ Equations
Mathematical Physics, Analysis and Geometry
Volume 3, Issue 2 , pp 139-177
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- hypergeometric solutions
- Kac–Moody Lie algebras
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- Author Affiliations
- 1. Departement Mathematik, ETH-Zentrum, 8092, Zürich, Switzerland
- 2. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599 – 3250, U.S.A.
- 3. St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011, Russia
- 4. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599 – 3250, U.S.A