Block type symmetry of bigraded Toda hierarchy

Abstract

In this paper, we define Orlov-Schulman's operators ML, MR, and then use them to construct the additional symmetries of the bigraded Toda hierarchy. 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 the literature. By acting on two different spaces under the weak W-constraints, we find in particular two representations of this Block type Lie algebra.