Interpolations between Jordanian Twists Induced by Coboundary Twists

Abstract

We propose a new generalisation of the Jordanian twist (building on the previous idea from [Meljanac S., Meljanac D., Pacho l A., Pikutić D., J. Phys. A: Math. Theor. 50 (2017), 265201, 11 pages]). Obtained this way, the family of the Jordanian twists allows for interpolation between two simple Jordanian twists. This new version of the twist provides an example of a new type of star product and the realization for noncommutative coordinates. Real forms of new Jordanian deformations are also discussed. Exponential formulae, used to obtain coproducts and star products, are presented with details.

One Parameter Family of Jordanian Twists

“…If F = F 0 is a simple Jordanian twist, and ω = ω R = exp − u κ P D , we obtain the twist [1]. This also shows that, for any u, twist F R,u satisfies the 2-cocycle and normalization conditions.…”

“…Any Drinfeld twist F can be modified by any 1-cochain ω ∈ H, producing a new twist ω −1 ⊗ ω −1 F∆(ω), see [27]. In an earlier paper [1], this procedure has been used to obtain a certain twist F −1 R,u for every u. In that context, it has been written in the form of a product of three exponential factors, see also reference [31].…”

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].…”

Noncommutative field theory from an angular twist

“…These include a study of many different aspects among which we mention various scenarios of variable speed of light (VSL) [6], [7], corrections to neutrino propagation [8], [9], different schemes for implementing a relativity theory with minimum length [10], [11], [12], [13], [14], the principle of relative locality [15] and time delay in arrival of photons with different energy [16], [17], [18], [19]. There are diverse theoretical frameworks [20], [21], [22], [23], [24], [25] based on noncommutative (NC) spacetime which give rise to modified dispersion relations.…”