The Baxter's Q-operator for the W-algebra $W_N$

TL;DR

This paper constructs the Baxter's Q-operator for the W-algebra $W_N$ using q-oscillator representations, providing explicit free field realizations and functional relations, including a higher-rank generalization of the T-Q relation.

Contribution

It introduces a novel q-oscillator representation approach to realize Baxter's Q-operator for $W_N$, extending the T-Q relations to higher ranks.

Findings

Explicit free field realizations of Q-operators for $W_N$

Functional relations including higher-rank T-Q relations

Extension of Baxter's relations to $W_N$ algebra

Abstract

The q-oscillator representation for the Borel subalgebra of the affine symmetry $U_q(sl_N^)$ is presented. By means of this q-oscillator representation, we give the free field realizations of the Baxter's Q-operator $Q_j(t)$, $\bar{Q}_j(t)$ for the W-algebra $W_N$. We give the functional relations of the $T$-$Q$ operators, including the higher-rank generalization of the Baxter's $T$-$Q$ relation.