Bulg. J. Phys. vol.33 no.3 (2006), pp. 169-181

Geometric Solutions of Upper Triangular Toda Hierarchies

G.F. Helminck1, M.G. Mishina2, S.V. Polenkova2
1Faculty of Mathematical Sciences, University of Twente, P.O. Box 217, 7500 AE Enschede, The
2Derzhavin Tambov State University, Internatsionalnaya 33, 392622 Tambov, Russia
Abstract. In this paper one considers commutative subalgebras of the Z×Z-matrices generated by a maximal commutative subalgebra of the complex k×k-matrices and a shift matrix that commutes with them. Key objects are parameter dependent perturbations of these algebras inside the upper triangular Z×Z-matrices such that the perturbed generators satisfy Lax equations with respect to an infinite number of commuting directions. Various appropriate geometric settings are described in which one can actually construct solutions of these hierarchies.

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