Block type symmetry of bigraded Toda Hierarchy

TL;DR

This paper introduces the Block type Lie algebra structure of additional symmetries in the bigraded Toda hierarchy, expanding understanding of its algebraic symmetries and their representations.

Contribution

It defines new Orlov-Schulman's operators and demonstrates that the hierarchy's symmetries form a Block type Lie algebra with specific representations.

Findings

Symmetries form a Block type Lie algebra.

Two representations of the algebra are constructed.

The algebraic structure enriches the symmetry analysis of BTH.

Abstract

In this paper, we define Orlov-Schulman's operators $M_L$, $M_R$, and then use them to construct the additional symmetries of the bigraded Toda hierarchy (BTH). We further show that these additional symmetries form an interesting infinite dimensional Lie algebra known as a Block type Lie algebra, whose structure theory and representation theory have recently received much attention in literature. By acting on two different spaces under the weak W-constraints we find in particular two representations of this Block type Lie algebra.