A Lax type operator for quantum finite W-algebras

TL;DR

This paper constructs a Lax operator within quantum finite W-algebras for reductive Lie algebras, demonstrating its satisfaction of a generalized Yangian identity in classical cases, and introduces a quantum analogue of a classical operator.

Contribution

It introduces a Lax operator for quantum finite W-algebras associated with reductive Lie algebras, extending classical Adler-type operators to the quantum setting.

Findings

Lax operator L(z) constructed for quantum finite W-algebras.

L(z) satisfies a generalized Yangian identity for classical Lie algebras.

L(z) is a quantum analogue of a classical Adler-type operator.

Abstract

For a reductive Lie algebra g, its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra W(g,f). We show that for the classical linear Lie algebras gl_N, sl_N, so_N and sp_N, the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant.