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

摘要

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