The extended bigraded Toda hierarchy

Abstract

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 ǫ-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy, generalizing [4]. 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 ˜ W (N) (AN+M−1) of the A series, defined in [9].