Lie bundle on the space of deformed skew-symmetric matrices

Abstract

We study a Lie algebra ${\mathcal{A}}_{a_1,\dots ,a_{n-1}}$ 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 $a_1,\dots ,a_{n-1}$. Using isomorphism ${\mathcal{A}}_{a_1,\dots ,a_{n-1}}^{\ast }\cong L_{+}$ 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.

References in corpus (4)

• Integrable Hamiltonian systems related to the Hilbert--Schmidt ideal
• On the Cases of Kirchhoff and Chaplygin of the Kirchhoff Equations
• Integrable Systems Related to Deformed $\mathfrak{so}(5)$
• Integrable relativistic systems given by Hamiltonians with momentum-spin-orbit coupling