Block type symmetry of bigraded Toda hierarchy

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. 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. C 2012 American Institute of Physics.

A Strongly Coupled Extended Toda Hierarchy and its Virasoro Symmetry

“…Among the algebraic structures of additional symmetries, the Virasoro symmetry is one kind of important symmetries such as [23]. In [24], we provided a kind of Block type algebraic structures for the bigraded Toda hierarchy (BTH) [14,16]. Later on, this Block type Lie algebra was found again in the dispersionless bigraded Toda hierarchy [17].…”

Multifold Darboux Transformations of the Extended Bigraded Toda Hierarchy

“…Meanwhile it was proved to govern Gromov-Witten invariant of the total descendent potential of P 1 orbifolds [15]. This hierarchy also lead to a series of results from analytical and algebraic considerations [16,17,18,19]. However the explicit solutions of the EBTH are still unknown because the constraint on spectral parameter is complicated and most of equations are nonlocal.…”

Block (or Hamiltonian) Lie Symmetry of Dispersionless D-Type Drinfeld–Sokolov Hierarchy

“…(29) can only take values in odd numbers, the algebra is a subalgebra of the Lie algebra considered in [16], which is a Block type Lie algebra as well as a special case of Hamiltonian type Lie algebras (see, e.g., [20,21]). The above results together with that on [19] …”

“…In [16], we provide a Block type algebraic structure of the bigraded Toda hierarchy (BTH) [17,18]. Later on, this Block type Lie algebra is found again in dispersionless bigraded Toda hierarchy [19].…”