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