Block algebra in two-component BKP and D type Drinfeld-Sokolov hierarchies

TL;DR

This paper constructs and analyzes a Block algebra of additional symmetries for two-component BKP and D type Drinfeld-Sokolov hierarchies, revealing their algebraic structures and relations to integrable systems.

Contribution

It introduces generalized additional symmetries forming a Block algebra for these hierarchies, highlighting their algebraic structure and connection to the bigraded Toda hierarchy.

Findings

Additional symmetry flows form a Block type algebra with modifications.

D type Drinfeld-Sokolov hierarchy has a complete Block type symmetry algebra.

The algebraic structure is similar to the bigraded Toda hierarchy.

Abstract

We construct generalized additional symmetries of a two-component BKP hierarchy defined by two pseudo-differential Lax operators. These additional symmetry flows form a Block type algebra with some modified(or additional) terms because of a B type reduction condition of this integrable hierarchy. Further we show that the D type Drinfeld-Sokolov hierarchy, which is a reduction of the two-component BKP hierarchy, possess a complete Block type additional symmetry algebra. That D type Drinfeld-Sokolov hierarchy has a similar algebraic structure as the bigraded Toda hierarchy which is a differential-discrete integrable system.