Commutative Families of the Elliptic Macdonald Operator

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.

References in corpus (3)

• Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications
• Kernel function and quantum algebras
• Elliptic Ding-Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator

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• On Elliptic Algebras and Large-n Supersymmetric Gauge Theories
• Elliptic Ding-Iohara Algebra and Commutative Families of the Elliptic Macdonald Operator
• Elliptic algebra, Frenkel-Kac construction and root of unity limit