The Extended Bigraded Toda hierarchy

Abstract

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups of the $A$ series.

32 pages, corrected typos

References in corpus (5)

• Quantum Affine Algebras and Deformations of the Virasoro and W-algebras
• The Topological CP^1 Model and the Large-N Matrix Integral
• Virasoro Symmetries of the Extended Toda Hierarchy
• The algebra of q-pseudodifferential symbols and the q-W_{KP}^n algebra
• The Hamiltonian structures of the two-dimensional Toda lattice and R-matrices

Cited by in corpus (36)

• Block type symmetry of bigraded Toda Hierarchy
• Tau function and Hirota bilinear equations for the Extended bigraded Toda Hierarchy
• The extended $Z_N$-Toda hierarchy
• Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials
• Dispersionless bigraded Toda Hierarchy and its additional symmetry
• Solutions of bigraded Toda hierarchy
• The multicomponent 2D Toda hierarchy: Discrete flows and string equations
• Block algebra in two-component BKP and D type Drinfeld-Sokolov hierarchies
• On the extended multi-component Toda hierarchy
• Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies
• Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of CP^1 orbifolds
• Fractional Volterra Hierarchy
• Two extensions of 1D Toda hierarchy
• Central Invariants of the Constrained KP Hierarchies
• Sato theory on the $q$-Toda hierarchy and its extension
• Additional symmetries of the extended bigraded Toda hierarchy
• On tau-functions for the Toda lattice hierarchy
• Classical $r$-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics
• On some classes of discrete polynomials and ordinary difference equations
• Local and non-local multiplicative Poisson vertex algebras and differential-difference equations
• Integrable structure of melting crystal model with two q-parameters
• Mirror symmetry for extended affine Weyl groups
• Integrable maps from Galois differential algebras, Borel transforms and number sequences
• Orbifold melting crystal models and reductions of Toda hierarchy
• On the squared eigenfunction symmetry of the Toda lattice hierarchy
• $E_8$ spectral curves
• Multi-fold Darboux transformations of the extended bigraded Toda hierarchy
• Constrained lattice-field hierarchies and Toda system with Block symmetry
• Extended lattice Gelfand-Dickey hierarchy
• Cubic Hodge integrals and integrable hierarchies of Volterra type
• Dressing operators in equivariant Gromov-Witten theory of ${\mathbb{CP}}^1$
• Darboux transformations and Fay identities for the extended bigraded Toda hierarchy
• Bosonic symmetries of the extended fermionic $(2N,2M)$-Toda hierarchy
• Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy
• Old and New Reductions of Dispersionless Toda Hierarchy
• A strongly coupled extended Toda hierarchy and its Virasoro symmetry