A construction of solutions of an integrable deformation of a commutative Lie algebra of skew hermitian $\mathbb{Z} \times \mathbb{Z} $-matrices

Author:

Aloysius Helminck, Gerardus Helminck, Paul Alexander Helminck

Keyword:

Nonlinear Sciences, Exactly Solvable and Integrable Systems, Exactly Solvable and Integrable Systems (nlin.SI)

journal:

date:

2023-10-30 16:00:00

Abstract

Inside the algebra $LT_{\mathbb{Z}}(R)$ of $\mathbb{Z} \times \mathbb{Z}$-matrices with coefficients from a commutative $\mathbb{C}$-algebra $R$ that have only a finite number of nonzero diagonals above the central diagonal, we consider a deformation of a commutative Lie algebra $\mathcal{C}_{sh}(\mathbb{C})$ of finite band skew hermitian matrices that is different from the Lie subalgebras that were deformed at the discrete KP hierarchy and its strict version. The evolution equations that the deformed generators of $\mathcal{C}_{sh}(\mathbb{C})$ have to satisfy are determined by the decomposition of $LT_{\mathbb{Z}}(R)$ in the direct sum of an algebra of lower triangular matrices and the finite band skew hermitian matrices. This yields then the $\mathcal{C}_{sh}(\mathbb{C})$-hierarchy. We show that the projections of a solution satisfy zero curvature relations and that it suffices to solve an associated Cauchy problem. Solutions of this type can be obtained by finding appropriate vectors in the $LT_{\mathbb{Z}}(R)$-module of oscillating matrices, the so-called wave matrices, that satisfy a set of equations in the oscillating matrices, called the linearization of the $\mathcal{C}_{sh}(\mathbb{C})$-hierarchy. Finally, a Hilbert Lie group will be introduced from which wave matrices for the $\mathcal{C}_{sh}(\mathbb{C})$-hierarchy are constructed. There is a real analogue of the $\mathcal{C}_{sh}(\mathbb{C})$-hierarchy called the $\mathcal{C}_{as}(\mathbb{R})$-hierarchy. It consists of a deformation of a commutative Lie algebra $\mathcal{C}_{as}(\mathbb{R})$ of anti-symmetric matrices. We will properly introduce it here too on the way and mention everywhere the corresponding result for this hierarchy, but we leave its proofs mostly to the reader.

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