A commutative algebra on degenerate $CP1$ and Macdonald polynomials

We introduce a unital associative algebra $A$ associated with degenerate $CP1$⁠. We show that $A$ is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby, we can regard $A$ as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using $A$ and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding–Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard–Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator $M(q,t1,t2)$ of Okounkov–Pandharipande [e-print arXiv:math-ph/0411210].