Title: Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic $\mathfrak{o}^{\star}(4)$ symmetries,
Abstract: We construct firstly the complete list of five quantum deformations of $D=4$
complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})\cong
\mathfrak{o}(3;\mathbb{C})\oplus \mathfrak{o}(3;\mathbb{C})$, describing
quantum rotational symmetry 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 $\mathfrak{o}(4;\mathbb{C})$:
Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian
$\mathfrak{o}(2,2)$ and quaternionic $\mathfrak{o}^{\star}(4)$. For
$\mathfrak{o}(3,1)$ 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
$\mathfrak{o}(4;\mathbb{C})$ we present new results.