Commutative Families of the Elliptic Macdonald Operator

TL;DR

This paper constructs commutative families of the elliptic Macdonald operator using elliptic algebraic structures, extending previous work on trigonometric cases and providing new algebraic frameworks.

Contribution

It introduces the use of elliptic Ding-Iohara-Miki and Feigin-Odesskii algebras to build commuting elliptic Macdonald operators, advancing the algebraic understanding of these operators.

Findings

Construction of commutative families of elliptic Macdonald operators.

Establishment of relations between elliptic operators and kernel functions.

Extension of algebraic frameworks from trigonometric to elliptic cases.

Abstract

In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages, arXiv:0904.2291], Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida constructed two families of commuting operators which contain the Macdonald operator (commutative families of the Macdonald operator). They used the Ding-Iohara-Miki algebra and the trigonometric Feigin-Odesskii algebra. In the previous paper [arXiv:1301.4912], the present author constructed the elliptic Ding-Iohara-Miki algebra and the free field realization of the elliptic Macdonald operator. In this paper, we show that by using the elliptic Ding-Iohara-Miki algebra and the elliptic Feigin-Odesskii algebra, we can construct commutative families of the elliptic Macdonald operator. In Appendix, we will show a relation between the elliptic Macdonald operator and its kernel function by the free field realization.