The Quantization of the Generalized mKdV Equations for $\hat {\SL}_2$

TL;DR

This paper develops quantum deformations of the integrals of motion for generalized mKdV equations associated with SL_2, utilizing vertex operator algebra and quantum Serre relations to construct a commutative family.

Contribution

It introduces a novel quantum deformation framework for the integrals of motion of generalized mKdV equations using vertex operator algebra techniques.

Findings

Constructed quantum deformations of integrals of motion.

Established quantum Serre relations for vertex operators.

Built a q-BGG resolution for the deformed algebra.

Abstract

We construct quantum deformations of the integrals of motion of the generalized mKdV equations for $\hat {\SL}_2$. For this, we give the relevant vertex operator algebra and prove quantum Serre relations for vertex operators, it allows to construct a $q$-BGG resolution and to deform the classical integrals of motion in a commutativ family.