Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models

TL;DR

This paper explores the elliptic Hall algebra's commutative subalgebras, deriving explicit formulas for generators and connecting them to integrable many-body systems and $q,t$-deformed matrix models.

Contribution

It extends the understanding of commutative rays in the elliptic Hall algebra, providing explicit formulas and linking to integrable systems and matrix models.

Findings

Explicit formulas for algebra generators in various representations.

Connection between commutative subalgebras and integrable many-body systems.

Discussion of $q,t$-deformation of associated matrix models.

Abstract

We extend our consideration of commutative subalgebras (rays) in different representations of the $W_{1+\infty}$ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$). Its advantage is that it possesses the Miki automorphism, which makes all commutative rays equivalent. Integrable systems associated with these rays become finite-difference and, apart from the trigonometric Ruijsenaars system not too much familiar. We concentrate on the simplest many-body and Fock representations, and derive explicit formulas for all generators of the elliptic Hall algebra $e_{n,m}$. In the one-body representation, they differ just by normalization from $z^nq^{m\hat D}$ of the $W_{1+\infty}$ Lie algebra, and, in the $N$-body case, they are non-trivially generalized to monomials of the Cherednik operators with action restricted to symmetric polynomials. In the Fock representation, the resulting operators are expressed through auxiliary polynomials of $n$ variables, which define weights in the residues formulas. We also discuss $q,t$-deformation of matrix models associated with constructed commutative subalgebras.