ИСТИНА

Complete classification of nonisomorphic (basic) quantum quasitriangular Hopf deformations of the complex orthogonal Lie algebra o(4;C) and of its real forms: Euclidean o(4), Lorentzian o(3,1), Kleinian o(2,2) and quaternionic o∗(4) Lie algebras is given in terms of classical r-matrices. All the r-matrices are skew-symmetric, and they satisfy homogeneous and nonhomogeneous classical Yang- Baxter equation. Using the isomorphisms o(4; C) ≃ o(3; C) ⊗ o(3; C), o(3; C) ≃ sl(2; C) we show that these classical r-matrices are multiparametric subordinated sums of standard and Jordanian classical r-matrices for sl(2;C) and its real forms su(2) and su(1,1) ≃ sl(2;R) and also of Abelian classical r-matrices for sl(2; C) ⊗ sl(2; C) and its real forms. Such structure of the basic classical r-matrices makes possible explicitly to find all basic quantum quasitriangular Hopf deformations of o(4; C) and of all its real forms: o(4), o(3, 1), o(2, 2) and o∗(4).