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+∞ algebra to the elliptic Hall algebra (or, equivalently, to the Ding-Iohara-Miki (DIM) algebra U q , t gl ̂ ̂ 1 $$ {U}_{q,t}\left({\hat{\hat{\mathfrak{gl}}}}_1\right) $$ ). 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 n q m D ̂ $$ {z}^n{q}^{m\hat{D}} $$ 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.