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Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
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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 $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$). 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 $z^nq^{m\hat D}$ 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.
Forward citations
Cited by 8 Pith papers
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Integrable systems inspired by DAHA and DIM algebra: type $C^\vee C$ versus type $A$
Type C∨C DAHA and Koornwinder systems mirror type-A Macdonald structures for Hamiltonians, recursions, evaluations and dualities, but lack a usable Noumi-Shiraishi-style universal series and SL(2,Z)-type twisting auto...
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Elliptic triad
The paper argues that the Macdonald triad admits elliptic deformations, but the defining linear equations for elliptic Baker-Akhiezer functions remain ambiguous except at m=1.
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Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems
In explicit small cases, twisted Baker-Akhiezer functions satisfy the defining linear equations and are eigenfunctions of the integer-ray DIM Hamiltonians.
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