Quantifying Algebraic Asymmetry of Hamiltonian Systems

The symmetries play important roles in physical systems. We study the symmetries of a Hamiltonian system by investigating the asymmetry of the Hamiltonian with respect to certain algebras. We define the asymmetry of an operator with respect to an algebraic basis in terms of their commutators. Detailed analysis is given to the Lie algebra $\mathfrak{s}\mathfrak{u}(2)$ and its $q$-deformation. The asymmetry of the $q$-deformed integrable spin chain models is calculated. The corresponding geometrical pictures with respect to such asymmetry is presented.