A Flagvariety Relating Matrix Hierarchies and Toda-Type Hierarchies

Abstract

Commutative subalgebras of the complex k × k-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 H 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. Mathematics Subject Classification (2000)17B80 · 37K10 · 58B25 · 81R12

An Algebraic Characterization of the Bilinear Relations of the Matrix Hierarchy Associated with a Commutative Algebra of k×k-Matrices

“…Such a convergent approach can be found e.g. in [9]. Here we present a more general algebraic approach.…”

“…The complex of (8), (9) and (11) are the basic relations of the h-hierarchy. The main interest is in (8) and (9), because they render the nonlinear differential equations, and they are called the Lax equations of the hierarchy.…”

“…The main interest is in (8) and (9), because they render the nonlinear differential equations, and they are called the Lax equations of the hierarchy. To give sense to this exercision, one has to make concrete realizations of the relations in (8), (9) and (11). This means that we search first of all for commutative C-algebras R equiped with a privileged derivation ∂ : R → R and a collection of derivations {∂ iα , i ≥ 1, 1 ≤ α ≤ n} of R that commute all with ∂ and also among each other.…”

An analytic framework for the two-dimensional infinite Toda hierarchy associated with a commutative algebra

“…It is a combination of two hierarchies, one formulated in terms of lower-triangular matrices and one formulated in terms of upper-triangular matrices. An analytic framework that incorporated the flows of the first hierarchy was treated in [3], and a geometric setting that includes the flows for the second type of hierarchy was given in [4]. The setup presented here allows both groups of flows and furnishes solutions of the combined hierarchy.…”

Аналитический Подход К Двумерной Бесконечной Иерархии Тоды, Связанной С Коммутативной Алгеброй

“…Она представляет собой комбинацию двух иерархий, одна из которых формулируется в терминах нижнетреугольных, а другая -в терминах верх-нетреугольных матриц. Аналитический подход к потокам первой иерархии рассмат-ривался в работе [3], а геометрический формализм для потоков иерархии второго типа приведен в работе [4]. В рамках формализма настоящей работы допускаются обе группы потоков и получаются решения для комбинированной иерархии.…”