Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Let H be a separable Hilbert space and T be a self-adjoint bounded linear operator on H-circle times 2 with norm <= 1, satisfying the Yang-Baxter equation. Bozejko and Speicher ([10]) proved that the operator T determines a T-deformed Fock space F(H) = circle plus F-infinity(n=8)n(H). We start with reviewing and extending the known results about the structure of the n-particle spaces .F-n (H) and the commutation relations satisfied by the corresponding creation and annihilation operators acting on F(H). We then choose H = L-2 (X -> V), the L-2-space of V-valued functions on X. Here X := R-d and V := C-m with m >= 2. Furthermore, we assume that the operator T acting on H-circle times 2 = L-2 (X-2 -> V-circle times 2) is given by (Tf-(2))(x, y) = C-x,C-y f((2))(y, x). Here, for a.a. (x, y) is an element of X-2, C-x,C-y is a linear operator on V-circle times 2 with norm <= 1 that satisfies C*(x,y) = C-y(,x) and the spectral quantum Yang-Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function C-xy in the case d = 2 determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its T-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.