Weyl Pair, Current Algebra and Shift Operator

TL;DR

This paper constructs a shift operator from Abelian current algebra on a lattice using Weyl pairs, revealing its relation to braid group operators and expressing it via theta functions or quantum dilogarithm depending on the parameter q.

Contribution

It introduces a novel realization of the shift operator within Abelian current algebra using Weyl pairs, connecting it to braid group representations and special functions.

Findings

Shift operator expressed as product of theta functions for |q|≠1

Shift operator expressed by quantum dilogarithm for |q|=1

Establishes a link between current algebra, Weyl pairs, and braid group operators

Abstract

The Abelian current algebra on the lattice is given from a series of the independent Weyl pairs and the shift operator is constructed by this algebra. So the realization of the operators of the braid group is obtained. For $|q|\neq 1$ the shift operator is the product of the theta functions of the generators $w_n$ of the current algebra. For $|q|=1$ it can be expressed by the quantum dilogarithm of $w_n$.