On pentagon identity in Ding-Iohara-Miki algebra

TL;DR

This paper reveals a unifying 'master pentagon identity' within the Ding-Iohara-Miki algebra that encompasses known identities for quantum dilogarithm functions and Macdonald polynomial operators, offering new insights into their interconnected structures.

Contribution

It introduces a general 'master pentagon identity' in the Ding-Iohara-Miki algebra that unifies previously known pentagon identities, expanding understanding of algebraic relations.

Findings

Verification of the master pentagon identity

Connections between quantum dilogarithm and Macdonald polynomials

Implications for algebraic structures and identities

Abstract

We notice that the famous pentagon identity for quantum dilogarithm functions and the five-term relation for certain operators related to Macdonald polynomials discovered by Garsia and Mellit can both be understood as specific cases of a general "master pentagon identity" for group-like elements in the Ding-Iohara-Miki (or quantum toroidal, or elliptic Hall) algebra. We perform some checks of this remarkable identity and discuss its implications.