2 v 1 3 0 Ja n 19 97 On the braided Fock spaces

Framework for constructing Fock spaces associated either with certain solutions of the quantum Yang-Baxter equation or with infinite dimensional Hecke algebra is presented. For the former case, the quantum deformed oscillator algebra associated with the solution of the quantum Yang-Baxter equation is found. During last decade much attention have been paid to the quantum (q-deformed) groups and algebras (both will be called quantum groups below) as well as to their applications in such diverse branches of theoretical and mathematical physics as conformal field theory, integrable models, nuclear physics, statistical mechanics, knot theory and topological field theory (see, e.g., [1] and references therein). For the first time they appeared in fact as the main ingredient of quantum inverse scattering method [2, 3] and then were interpreted as Hopf algebras in [4, 5, 6] (see also [7] for the quantum matrix group interpretation). There exist various models describing systems of either one particle or several distinct particles, which possess quantum symmetry, i.e., are symmetrical with respect to the action of quantum group (see, e.g., [1], [8]-[10]). Unfortunately, the problem of description of a quantum symmetric (identical) multiparticle system is not solved completely yet. It is well known that quantum groups are deeply connected with the braid group, which substitutes usual symmetric group in this case [11, 12, 13] (see also [14]). Therefore, one can expect that to construct a multiparticle system possessing quantum symmetry, the braid group should be used instead of symmetric group. On the other hand, it is known that inequivalent quantizations of multiparticle systems on two-dimensional manifolds are labeled by the irreducible representations of the braid group, giving rise to braid statistics [15, 16, 17]. Being motivated by above-mentioned arguments, in this paper we construct the Fock spaces of particles obeying braid statistics. Recent state of the problem is as follows. The Fock spaces of the particles obeying statistics associated with nontrivial representations of the symmetric group have been constructed in [18]. The same has been done in [19] for the multiparticle systems with the special emphasis on the quantum symmetrical properties of systems under consideration. In the number of papers Fock spaces are constructed, which originate from the creation and annihilation operators with nontrivial commutation relations being preserved under the action of some quantum groups [20] (see also [21]-[24]). The realization of the q-deformed Fock space in terms of q-wedges have been done in [25]. In the present …