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 $U_{q,t}({\widehat{\hat{\mathfrak{gl}}}}_1)$). 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 $z^n{q}^{m\hat{D}}$ 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.

arXiv comment51 pages, LaTeX

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Cited by in corpus (12)

• On Chalykh's approach to eigenfunctions of DIM-induced integrable Hamiltonians
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• Twisted Cherednik spectrum as a $q,t$-deformation
• Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems
• Phases in WLZZ Matrix Models
• Diamond of triads
• Super-Hamiltonians for super-Macdonald polynomials
• Elliptic triad
• $W_{1+\infty }$ flows and multi-component hierarchy (KP case)
• Generating twisted Cherednik eigenfunctions
• CMM formula as superintegrability property of unitary model
• Twisted Baker-Akhiezer function from determinants