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.

51 pages, LaTeX

References in corpus (21)

• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• Superintegrability summary
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Interpolating Matrix Models for WLZZ series
• On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations
• Many-body integrable systems implied by WLZZ models
• New Insights into Superintegrability from Unitary Matrix Models
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• Commutative subalgebras from Serre relations
• Duality in elliptic Ruijsenaars system and elliptic symmetric functions
• Quantum toroidal algebras and solvable structures in gauge/string theory
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Superintegrability as the hidden origin of Nekrasov calculus
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• Elliptic $q,t$ matrix models
• Towards elliptic deformation of $q,t$-matrix models
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Summing up perturbation series around superintegrable point
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model

Cited by in corpus (12)

• On Chalykh's approach to eigenfunctions of DIM-induced integrable Hamiltonians
• Supersymmetric polynomials and algebro-combinatorial duality
• 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