A Flagvariety Relating Matrix Hierarchies and Toda-Type Hierarchies

G.F. Helminck

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    2 Citations (Scopus)


    Commutative subalgebras of the complex -matrices are known to generate both matrix and Toda-type hierarchies. In this paper a certain class of infinite chains of closed subspaces of a separable Hilbert space will be introduced. To each such a flag one associates a sequence of solutions of the matrix hierarchy related to this subalgebra. They compose to a solution of the lower triangular Toda hierarchy corresponding to the transposed algebra. Both solutions can be expressed in determinants of suitable Fredholm operators, the so-called τ-functions. These last functions also have a geometric interpretation in terms of line bundles over the flagvariety. They measure the failure of equivariance w.r.t. to the commuting flows of certain global sections.
    Original languageUndefined
    Pages (from-to)121-142
    JournalActa applicandae mathematicae
    Issue number1-2
    Publication statusPublished - 2006


    • IR-69567
    • Flag variety - line bundle - commuting flows - matrix hierarchies - Toda-type hierarchies - τ-functions

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