We discuss relations between different integral formulae for solutions of the quantized Knizhnik-Zamolodchikov (qKZ) equation at level zero in the Uq(sl2) case for |q | < 1 . Smirnov type formulae of M. Jimbo et al. are derived from the general approach of A.Varchenko and the author. The consideration is parallel to the qKZ equation in the rational sl2 case done by A.Nakayashiki, S. Pakuliak an...

It is shown that the celebrated Menelaus relation, Hirota–Miwa bilinear equation for KP hierarchy and Fay&apos;s trisecant formula similar to WDVV are associativity conditions structure constants of certain three-dimensional quasi-algebra.

We show explicitly a generalised Lie algebra embedded in the positive and negative parts of the Drinfeld-Jimbo quantum groups of type An. Such a generalised Lie algebra satisfy axioms closely related to the ones found by S.L. Woronowicz. For the universal enveloping algebra of such generalised Lie algebras we establish several conditions in order to obtain bases of type Poincaré-Birkhoff-Witt. ...

A theoretical foundation for a generalization of the elliptic difference Painlevé equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transforma...

We show that for Jack parameter α = −(k+1)/(r−1), certain Jack polynomials studied by Feigin–Jimbo–Miwa–Mukhin vanish to order r when k + 1 of the coordinates coincide. This result was conjectured by Bernevig and Haldane, who proposed that these Jack polynomials are model wavefunctions for fractional quantum Hall states. Special cases of these Jack polynomials include the wavefunctions of Laugh...

We derive a Toda-type recurrence relation, in both highand low-temperature regimes, for the λ-extended diagonal correlation functions C(N,N;λ) of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal co...