Addendum to “Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o⋆(4) symmetries in unified o(4;C) setting” [Phys. Lett. B 754 (2016) 176–181]

Abstract

In our previous paper [1] we obtained a full classification of nonequivalent quasitriangular quantum deformations for the complex D=4 Euclidean Lie symmetry o(4;C). The result was presented in the form of a list consisting of three three-parameter, one two-parameter and one one-parameter nonisomorphic classical r-matrices which provide ‘directions’ of the nonequivalent quantizations of o(4;C). Applying reality conditions to the complex o(4;C) r-matrices we obtained the nonisomorphic classical r-matrices for all possible real forms of o(4;C): Euclidean o(4), Lorentz o(3,1), Kleinian o(2,2) and quaternionic o⋆(4) Lie algebras. In the case of o(4) and o(3,1) real symmetries these r-matrices give the full classifications of the inequivalent quasitriangular quantum deformations, however for o(2,2) and o⋆(4) the classifications are not full. In this paper we complete these classifications by adding three new three-parameter o(2,2)-real r-matrices and one new three-parameter o⋆(4)-real r-matrix. All nonisomorphic classical r-matrices for all real forms of o(4;C) are presented in the explicit form what is convenient for providing the quantizations. We will mention also some applications of our results to the deformations of space–time symmetries and string σ-models.