We give a new presentation of the Drinfeld double E of the (spherical) elliptic Hall algebra E+ introduced in our previous work (Burban and Schiffmann in Duke Math. J. preprint math.AG/0505148, 2005). This presentation is similar in spirit to Drinfeld’s ‘new realization’ of quantum affine algebras. This answers, in the case of elliptic curves, a question of Kapranov concerning functional relati...

Functional equations, in the form of fusion hierarchies, are studied for the transfer matrices of the fused restricted A (1) n−1 lattice models of Jimbo, Miwa and Okado. Specifically, these equations are solved analytically for the finite-size scaling spectra, central charges and some conformal weights. The results are obtained in terms of Rogers dilogarithm and correspond to coset conformal fi...

The aim of this paper is to generalize several aspects of the recent work of LeclercThibon and Varagnolo-Vasserot on the canonical bases of the level 1 q-deformed Fock spaces due to Hayashi. Namely, we define canonical bases for the higher-level qdeformed Fock spaces of Jimbo-Misra-Miwa-Okado and establish a relation between these bases and (parabolic) Kazhdan-Lusztig polynomials for the affine...

In this paper, we construct Grammian-like quasideterminant solutions of a non-Abelian Hirota–Miwa equation. Through continuum limits of this non-Abelian Hirota–Miwa equation and its quasideterminant solutions, we construct a cascade of noncommutative differential-difference equations ending with the non-commutative KP equation. For each of these systems, the quasideterminant solutions are const...

A non-Abelian version of the Hirota-Miwa equation is considered. In an earlier paper [Nimmo (2006) J. Phys. A: Math. Gen. 39, 5053-5065] it was shown how solutions expressed as quasideterminants could be constructed for this system by means of Darboux transformations. In this paper we discuss these solutions from a different perspective and show that the solutions are quasi-Plücker coordinates ...

We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system, to its Lax pair. Thus we obtain new 3×3 matrix Fuchs–Garnier pairs for the third and fifth Painlevé equations, together with the previously known Fuchs– Garnier pair for the fourth and sixth Painlevé equations. These Fuchs–Garnier pairs have an important feature: they are linear with respect...

Article history: Received 2 September 2008 Received in revised form 8 October 2008 Accepted 8 October 2008 Available online 11 October 2008 Communicated by R. Wu PACS: 02.30.Jr 04.20.Jb 05.45.Yv

The polynomial ring Br:=Q[e1,…,er] in r indeterminates is a representation of the Lie algebra all endomorphism Q[X] vanishing at Xj for but finitely many j. We determine structural formal power series Br-representation gl∞(Q). This r+2 encoding images basis elements Br under action generating function elementary endomorphisms Q[X]. obtained expression implies (and improves) formula by Gatto & S...