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 $$ {U}_{q,t}\left({\hat{\hat{\mathfrak{gl}}}}_1\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mfenced> <mml:msub> <mml:mover> <mml:mover> <mml:mi>gl</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:mfenced> </mml:math> ). 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 $$ {z}^n{q}^{m\hat{D}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:mrow> </mml:msup> </mml:math> 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.