Lie bundle on the space of deformed skew-symmetric matrices

TL;DR

This paper introduces a new class of Lie algebras formed by deformed skew-symmetric matrices, establishing a Lie-Poisson structure on upper-triangular matrices and generating integrable Hamiltonian hierarchies with bihamiltonian structure.

Contribution

It defines a novel family of Lie algebras parametrized by deformation parameters and constructs associated Hamiltonian hierarchies with bihamiltonian structures.

Findings

Established isomorphism between the Lie algebra and upper-triangular matrices

Constructed Lie-Poisson structures on matrix spaces

Generated integrable Hamiltonian hierarchies with bihamiltonian properties

Abstract

We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real numbers $a_1,\ldots,a_{n-1}$. Using isomorphism $\mathcal A_{a_1,\ldots,a_{n-1}}^* \cong L_+$ we introduce a Lie-Poisson structure on the space of upper-triangular matrices $L_+$. In this way we generate hierarchies of Hamilton systems with bihamiltonian structure.