Year: 2024
Type: article
Abstract: 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 $$ {U}_{q,t}\left({\hat{\hat{\mathfrak{gl}}}}_1\right) $$ U q , t 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 $$ {z}^n{q}^{m\hat{D}} $$ z n q m 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.
Source: Journal of High Energy Physics
Authors: A. Mironov, A. Morozov, A. Popolitov
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: Russian Science Foundation
Award: G6822810008
OpenAlex ID: W4402915058