Quantum Knizhnik-Zamolodchikov equations and affine root systems. (English) Zbl 0849.17025

Summary: Quantum (difference) Knizhnik-Zamolodchikov (QKZ) equations [F. A. Smirnov, J. Phys. A 19, L 575–L 578 (1986; Zbl 0617.58048); I. B. Frenkel and N. Yu. Reshetikhin, Commun. Math. Phys. 146, No. 1, 1-60 (1992; Zbl 0760.17006)] are generalized for the \(R\)-matrices from the author’s note [ICM-90 Satell. Conf. Proc., 63-77 (1991; Zbl 0780.17032)] with the arguments in arbitrary root systems (and their formal counterparts). In particular, QKZ equations with certain boundary conditions are introduced. The self-consistency of the equations from Frenkel and Reshetikhin, and the cross-derivative integrability conditions for the \(r\)-matrix KZ equations from [Sov. Math., Dokl. 40, 43-48 (1990); translation from Dokl. Akad. Nauk SSSR 307, No. 1, 49-53 (1989; Zbl 0747.17017)] are obtained as corollaries. A difference counterpart of the quantum many-body problem connected with Macdonald’s operators is defined as an application.


17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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