Once again about quantum deformations of D=4 Lorentz algebra: twistings of q-deformation

TL;DR

This paper explores advanced quantum deformations of the D=4 Lorentz algebra, focusing on twistings of the q-deformation, and provides explicit formulas for the resulting Hopf algebra structures.

Contribution

It introduces new twist deformations of the D=4 Lorentz algebra, including Abelian, Cremmer-Gervais, and Jordanian twists, with explicit coproduct and antipode formulas.

Findings

Derived explicit formulas for deformed coproducts and antipodes.

Presented two new quantum deformations via twisting techniques.

Extended the understanding of quantum Lorentz algebra structures.

Abstract

This paper together with the previous one (arXiv:hep-th/0604146) presents the detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation an Abelian twist depending on Cartan generators of o(3,1) is used. The second example of twisting provides a quantum deformation of Cremmer-Gervais type for the Lorentz algebra. For completeness we describe also twisting of the Lorentz algebra by standard Jordanian twist. By twist quantization techniques we obtain for these deformations new explicit formulae for the deformed coproducts and antipodes of the o(3,1)-generators.