One Parameter Family of Jordanian Twists

Abstract

We propose an explicit generalization of the Jordanian twist proposed in rsymmetrized form by Giaquinto and Zhang. It is proved that this generalization satisfies the 2-cocycle condition. We present explicit formulas for the corresponding star product and twisted coproduct. Finally, we show that our generalization coincides with the twist obtained from the simple Jordanian twist by twisting by a 1-cochain.

Interpolations between Jordanian twists, the Poincaré–Weyl algebra and dispersion relations

“…For u = 1/2, F L,u=1/2 corresponds to the twist proposed in [9]. For u = 1, it follows from in [16] and [17] that F L,u=1 is identical to the Jordanian twist…”

“…Dilatation D is included in a minimal extension of the relativistic space time symmetry, the so-called Poincaré-Weyl algebra generated by {M µν , p µ , D}, where M µν denote the Lorentz generators. One parameter interpolations between Jordanian twists, which are generated from a simple Jordanian twist F 0 by twisting with 1-cochains, were studied in [15,16,17].…”

“…For u = 0, F R,u=0 reduces to the Jordanian twist F 0 and for u = 1 it was shown in [17] that F R,u=1 = F 1 . Hence, the family of twists F R,u interpolates between two Jordanian twists F 0 and F 1 .…”

“…Hence, the twist F R,u=1/2 is given by (11) and satisfies the normalization and cocycle condition automatically, as it is obtained as the cochain twist of a normalized 2-cocycle. It was shown in [17] that the twist F −1 R,u can be written as…”

Progress in Snyder model

“…where the function χ(β p 2 ) is arbitrary and does not affect the commutation relations, but takes into account ambiguities arising from operator ordering of ξ µ and p µ . In general, it can be shown [39] that for any noncommutative model, e ik•x e iq•ξ = e iP(k,q)•ξ +iQ(k,q) , (5.3)…”