On representations of Lie algebras of a generalized Tavis-Cummings model

Abstract

Consider the Lie algebras $L_{r,t}^s:\left[K_1,K_2\right]=s{K}_3$, $\left[K_3,K_1\right]=r{K}_1$, $\left[K_3,K_2\right]=-r{K}_2$, $\left[K_3,K_4\right]=0$, $\left[K_4,K_1\right]=-t{K}_1$, and $\left[K_4,K_2\right]=t{K}_2$, subject to the physical conditions, $K_3$ and $K_4$ are real diagonal operators representing energy, $K_2=K_1^{†}$, and the Hamiltonian $H={\omega }_1{K}_3+\left({\omega }_1+{\omega }_2\right)K_4+\lambda \left(t\right)\left(K_1{e}^{i\Phi }+K_2{e}^{i\Phi }\right)$ is a Hermitian operator. Matrix representations are discussed and faithful representations of least degree for $L_{r,t}^s$ satisfying the physical requirements are given for appropriate values of $r,s,t\in ℝ$.