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]

摘要

In our previous paper 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.