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

A bstract We extend our consideration of commutative subalgebras (rays) in different representations of the W 1+ ∞ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra Uₐ, ₓ ({{gl}}₁) U q, t 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 zⁿq^mD z n q m D ̂ of the W 1+ ∞ 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.