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.
Highlights
The search for quantum gravity is linked with studies of noncommutative space-times and quantum deformations of space-time symmetries
In this paper we provide the full classifications of the quantum deformations in the case of Kleinian o(2, 2) and quaternionic o⋆(4) symmetries by adding to the results in [1] three new three-parameter o(2, 2)-real r-matrices and one new three-parameter o⋆(4)-real r-matrix
If we consider a Lie algebra over R with the commutation relations (2.5) we get the compact real form o(3) := o(3; R) with the anti-Hermitian basis (i = 1, 2, 3): Ii∗ = −Ii for o(3)
Summary
The search for quantum gravity is linked with studies of noncommutative space-times and quantum deformations of space-time symmetries. Applying reality conditions in [1] 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.
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