Abstract
We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra mathfrak{o}left(4;mathbb{C}right)cong mathfrak{o}left(3;mathbb{C}right)oplus mathfrak{o}left(3;mathbb{C}right) , describing quantum rotational symmetries of four-dimensional complex space-time, in particular we provide the corresponding universal quantum R-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of mathfrak{o}left(4;mathbb{C}right) : Euclidean mathfrak{o}(4) , Lorentz mathfrak{o}left(3, 1right) , Kleinian mathfrak{o}left(2, 2right) and quaternionic {mathfrak{o}}^{star }(4) . For mathfrak{o}left(3, 1right) we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra mathfrak{o}left(4;mathbb{C}right) we present new results.
Highlights
We construct firstly the complete list of five quantum deformations of D = 4 complex homogeneous orthogonal Lie algebra o(4; C) ∼= o(3; C) ⊕ o(3; C), describing quantum rotational symmetries of four-dimensional complex space-time, in particular we provide the corresponding universal quantum R-matrices
In this paper we presented the complete set of Hopf-algebraic quantum deformations generated by classical r-matrices for o(4; C) and its real forms given in [20, 22]
We provide the explicit formulae describing algebraic and coalgebraic sectors as well as the universal R-matrices which describe the tensoring of quantum modules
Summary
It is known that formulated by Drinfeld [10, 11] the quantization problem of Lie bialgebras has been answered by Etingof and Kazhdan [14]: to each Lie bialgebra one can associate a quantized enveloping algebra supplemented with Hopf algebra structure (see [15] for less technical presentation). [15,16,17]): A) If in (2.1) t = 0, one gets the so-called triangular or non-standard case with vanishing Schouten brackets describing homogenous classical Yang-Baxter equation (denoted as CYBE). B) If t = 0, eq (2.1) describes so-called non-triangular (quasitriangular) classical r-matrix, satisfying inhomogenous or modified classical Yang-Baxter equation (mCYBE) In such case one can introduce rBD ∈ g ⊗ g called Belavin-Drinfeld form of the r-matrix satisfying CYBE, such that r = rBD − rBτ D ((x ⊗ y)τ = y ⊗ x is the flip operation) and the symmetric element rBs D ≡ rBD + rBτ D which is ad-invariant.. We would like to add that the skew-symmetric classical r-matrices are sufficient for classification as well as for the description of the correspondence with classical Lie-Poisson groups
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