Isomorphisms between Quantum Group Covariant q - Oscillator Systems Defined for q and q − 1

It is shown that there exists an isomorphism between q-oscillator systems covariant under SUq(n) and SUq−1(n). By the isomorphism, the defining relations of SUq−1(n) covariant q-oscillator system are transmuted into those of SUq(n). It is also shown that the similar isomorphism exists for the system of q-oscillators covariant under the quantum supergroup SUq(n/m). Furthermore the cases of q-deformed Lie (super)algebras constructed from covariant q-oscillator systems are considered. The isomorphisms between q-deformed Lie (super)algebras can not obtained by the direct generalization of the one for covariant q-oscillator systems. PACS 02.20.Tw, 02.20.Qs Address after April 1994, Department of Applied Mathematics, Osaka Women’s University, Sakai, Osaka 590, Japan 1 Since the discovery of quantum deformation (the so-called q-deformation) of Lie groups and Lie algebras [1-5], many q-deformed objects have been introduced. We can mention q-deformed hyperplane [6], differential forms and derivatives on q-deformed hyperplane [7], q-(super)oscillators [8, 9], q-deformed covariant oscillator systems [10-13] and their generalization [14], q-symplecton [12, 15], reflection equation algebras [16], and so on. Almost all of these objects are essentially defined by the same algebraic structure, that is, Zamolodchikov-Faddeev algebra [17] or quantum group tensor [18]. However the relationship between q-deformed objects defined for different values of the deformation parameter q is unclear. This problem has been discussed for the q-oscillator Hq = {a, a , N} and found that the central element of Hq plays a crucial role. Assuming that the element N and the central element are independent of q, Chaichian et al. derived the formula which transforms the elements of Hq1 to the corresponding ones of Hq2 [19]. Without such assumption, the present author found the one-to-one correspondence between the elements of Hq and Hq−1 which transmute the defining relations of Hq−1 into those of Hq [20]. The elements of Hq−1 can be expressed in terms of those of Hq, therefore, we can say that the algebra Hq is invariant under the replacement q ↔ q . In mathematical language, Hq is isomorphic to Hq−1. In this article, it is shown that there exists an isomorphism between q-oscillator systems which are covariant under SUq(n) and SUq−1(n). By the isomorphism, the defining relations of SUq−1(n) covariant q-oscillator system are transmuted into those of SUq(n). It is also shown that the similar isomorphism exists for the system of q-oscillators covariant under the quantum supergroup SUq(n/m). Furthermore q-deformed Lie (super)algebras constructed from covariant q-oscillator systems are considered. They are also covariant under the coaction of SUq(n) and SUq(n/m) . It is shown that, unfortunately, the isomorphisms between covariant q-oscillator systems are not applicable to establish the isomorphisms between q-deformed Lie (super)algebras. We start with the SUq(n) covariant q-oscillator system Aq. It is generated by 2n generators {Ai, A † i , i = 1, · · ·n} and they satisfy the following defining relations [10, 11, 12] AiAj = qAjAi, A † iA † j = q A†jA † i , i < j AiA † j = qA † jAi, (1) AiA † i − q A†iAi = 1 + (q 2 − 1) i−1