Lie superbialgebra structures on the Lie superalgebra $({\cal C}^3 + {\cal A})$ and deformation of related integrable Hamiltonian systems

TL;DR

This paper classifies Lie superbialgebra structures on a specific Lie superalgebra, explores their quantum deformations, and constructs a deformed integrable Hamiltonian system based on these structures.

Contribution

It provides a complete classification of Lie superbialgebra structures on $({ m C}^3 + { m A})$, determines their quantum deformations, and applies these to develop a deformed integrable Hamiltonian system.

Findings

31 inequivalent Lie superbialgebra families identified.

Explicit quantum deformations and deformed Casimir elements obtained.

Construction of a deformed integrable Hamiltonian system from the quantum algebra.

Abstract

Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra $({\cal C}^3 + {\cal A})$ are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation. Then, by making use of the automorphism supergroup of the Lie superalgebra $({\cal C}^3 + {\cal A})$, the Lie superbialgebra structures on the Lie superalgebra $({\cal C}^3 + {\cal A})$ are obtained and classified into inequivalent 31 families. We also determine all corresponding coboundary and bi-r-matrix Lie superbialgebras. The quantum deformations associated with some Lie superbialgebras $({\cal C}^3 + {\cal A})$ are obtained, together with the corresponding deformed Casimir elements. As an application of these quantum deformations, we construct a deformed integrable Hamiltonian system from the representation of the Hopf superalgebra ${{U}_{_\lambda}}^{\hspace{-1mm}({\cal C}_{p=1}^{2,\epsilon} \oplus {\cal A}_{1,1})}\big(({\cal C}^3+{\cal A})\big)$.