Deformed boson algebras and the quantum double construction

The quantum double construction of a q-deformed boson algebra possessing a Hopf algebra structure is carried out explicitly. The R-matrix thus obtained is compared with the existing literature. Recently there has been an increasing interest in the deformation of Lie (super)algebras[1, 2, 3, 4, 5, 6] and their quasitriangular Hopf algebra nature[7], mainly because of there wide applications in mathematical physics. Parallel attempts to consistently q-deform the boson algebra also appeared[8, 9, 10, 11, 12, 13, 14, 15, 16] both independently and in connection with quantum group realizations, addressing also their possible Hopf algebra nature[17, 18]. The main aim of this letter is, on the one hand to point out the ambigious validity of an R-matrix obtained from a definition of q-boson algebra endowed with a Hopf algebra structure[17], and on the other to demonstrate the quantum double construction[1, 19, 20] for this algebra which will lead to an unambiguously valid R-matrix. The q-deformed boson algebras, denoted here by L, that have been considered are usually taken to be generated by a, a and N subject to the following commutation relations: [N, a] = −a, [N, a] = a, (1) together with one out of the following list of additional relations: [a, a] = [N + I]q − [N ]q , (2) aa − qaa = q , (3) aa − qaa = q , (4) aa = [N ], and aa = [N + I], (5) where I is the unit of L and as usual [x] = (qx − q−x)/(q − q−1), and q not a root of unity. When q = 1 we obtain the well known defining relations of the undeformed boson algebra. It should be mentioned that the consistency of the above definitions is justified as they can also be obtained from slq(2) by contraction[13, 14, 21]. Generalizations of q– boson defining relations, in particular that of (3), (4) have also been studied[9, 22, 23, 24]. Analysis of representations of L is quite rich [25, 24], but the most ususally used is the q–Fock representation (which has been shown [27] to be isomorphic with the usual boson Fock space by expressing the q–bosons as suitable functions of the undeformed bosons) given by: |n >= ([n]!)(a)|0 >, N |n >= n|n >, a|n >= [n+ 1]|n+ 1 >, a|n >= [n]|n− 1 > (6) where n = 0, 1, .... Using this representation, one can also show [21, 27, 18] the equivalence amongst the above definitions, which does not imply, though, an equivalence at the abstract algebraic level (as has been demonstrated in [18]). The most important point though concerns the Hopf algebra structure of the deformed boson algebra. Initially Hong Yan[17] showed that when L is defined by (1) and (2) (with N→N − 1/2, see (7) below) L is a Hopf algebra. Later this result was generalized in [18] where (1) was also generalized. We shall concentrate hereafter on the Hopf algebra L as defined in [17] by (1) and a symmetrized version of (2), namely [a, a] = [