Commutative Families of the Elliptic Macdonald Operator

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2014
• Main Author: Saito, Y.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2014
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/146833
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Commutative Families of the Elliptic Macdonald Operator / Y. Saito // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 8 назв. — англ.

In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages], 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.