Representations of sl(2,C) in the BGG category O and master symmetries

TL;DR

This paper explores the representation theory of sl(2,C) within the BGG category O and applies this to construct master symmetries for integrable nonlinear evolutionary systems, enabling the derivation of hierarchies of symmetries and conserved densities.

Contribution

It introduces a novel approach linking sl(2,C) modules in category O to the construction of master symmetries in integrable systems, including new equations.

Findings

Constructed master symmetries for classical and new integrable equations.

Generated conserved densities using master symmetries for novel equations.

Compared new method with existing approaches for symmetry construction.

Abstract

In this paper, we first give a short account on the indecomposable sl(2,C) modules in the Bernstein-Gelfand-Gelfand (BGG) category O. We show these modules naturally arise for homogeneous integrable nonlinear evolutionary systems. We then develop an approach to construct master symmetries for such integrable systems. This method naturally enables us to compute the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For the new integrable equations such as a Benjamin-Ono type equation, a new integrable Davey-Stewartson type equation and two different versions of (2+1)-dimensional generalised Volterra Chains, we generate their conserved densities using their master symmetries.