ON VECTOR–PARAMETER FORM OF THE $SU(2) \rightarrow SO(3,\mathbb{R})$ MAP

Abstract

By making use of the $\textit{Cayley}$ maps for the isomorphic Lie algebras $\mathfrak{su}(2)\text{ and }\mathfrak{so}(3)$ we have found the vector parameter form of the well-known $\textit{Wigner}\,$ group homomorphism $W : SU(2) \rightarrow SO(3,\mathbb{R})$ and its sections. Based on it and pulling back the group multiplication in $SO(3,\mathbb{R})$ through the $\textit{Cayley}$ map $\mathfrak{su}(2) \rightarrow SU(2)$ to the covering space, we present the derivation of the explicit formulas for compound rotations. It is shown that both sections are compatible with the group multiplications in $SO(3,\mathbb{R})$ up to a sign and this allows uniform operations with half-turns in the three-dimensional space. The vector parametrization of $SU(2)$ is compared with that of $SO(3,\mathbb{R})$ generated by the $\textit{Gibbs}$ vectors in order to discuss their advantages and disadvantages.