ИСТИНА

In this talk I would like to present complete classification of the classical $r$-matrices for the complex $D=4$ Euclidean algebra $\mathfrak{o}(4;\mathbb{C})$ and its real forms: Euclidian, quartenionic, Kleinian and Lorenzian. Moreover since the Lie algebra $\mathfrak{o}(4;\mathbb{C})$ is the direct sum, $\mathfrak{o}(4;\mathbb{C})=\mathfrak{o}(3;\mathbb{C})\oplus\bar{\mathfrak{o}}(3;\mathbb{C})\cong\mathfrak{sl}(2;\mathbb{C})\oplus\bar{\mathfrak{sl}}(2;\mathbb{C})$ therefore for the real forms $\mathfrak{o}(4;\mathbb{C})$ what we need to know is information about real forms of $\mathfrak{o}(3;\mathbb{C})\cong\mathfrak{sl}(2;\mathbb{C})$: $\mathfrak{o}(3)\cong\mathfrak{su}(2)$, $\mathfrak{o}(2,1)\cong\mathfrak{sl}(2;\mathbb{R})\cong\mathfrak{su}(1,1)$. The clasification of the classical $r$-matrices for $\mathfrak{su}(2)$ is well known. Concerning the case $\mathfrak{sl}(2;\mathbb{R})$ it is known already from 2000 (see X.Gomez ({\it JMP}, \textbf{41}, 7 (2000) 4939)). For the non-compact Lie algebrs $\mathfrak{su}(1,1)$ the total classification of their classical $r$-matrices is absent in literature. Here this classification will be obtained. For classification of the $\mathfrak{o}(4;\mathbb{C})$ classical $r$-matrices we shall use $\mathfrak{sl}(2;\mathbb{C})$-graded structure of the $r$-matrices and algebraic structure of Cartan--Weyl bases of our symmetries.