On Faithful Matrix Representations of q-Deformed Models in Quantum Optics

Consider the q-deformed Lie algebra, tq: [(K) over cap (1),(K) over cap (2]q)= (1-q)(K) over cap (1)(K) over cap (2),[(K) over cap (3),(K) over cap (1)](q)=s (K) over cap (3), [(K) over cap (1),(K) over cap (4]q)=s (K) over cap (4),[(K) over cap (3),(K) over cap (2)](q)=t (K) over cap (3),[(K) over cap (2),(K) over cap (4)](q)=t (K) over cap (4), and [(K) over cap (4),(K) over cap (3)]q = r (K) over cap1, where r,s,t is an element of R-{0}, subject to the physical properties: (K) over cap1 and (K) over cap2 are real diagonal operators, and (K) over cap (3)=(K) over cap (4)dagger, (dagger is for Hermitian conjugation). The q-deformed Lie algebra, t(q) is introduced as a generalized model of the Tavis-Cummings model (Tavis and Cummings 1968, Bashir and Sebawe Abdalla 1995), namely, [(K) over cap (1),(K) over cap](2)=0,[(K) over cap (1),(K) over cap](3)=-2 (K) over cap (3),[(K) over cap (1),(K) over cap (4])=2 (K) over cap (4),[(K) over cap (2),(K) over cap (3])=(K) over cap (3),[(K) over cap (2),(K) over cap (4)]=(K) over cap (4), and [(K) over cap (4),(K) over cap (3)] = (K) over cap (1), which is subject to the physical properties (K) over cap (1) and (K) over cap (2) are real diagonal operators, and (K) over cap (3)=(K) over cap (dagger)(4). Faithful matrix representations of the least degree of t(q) are discussed, and conditions are given to guarantee the existence of the faithful representations.