Embeddings of the Euclidean algebra $\mathfrak {e}(3)$e(3) into $\mathfrak {sl}(4,\mathbb {C})$sl(4,C) and restrictions of irreducible representations of $\mathfrak {sl}(4,\mathbb {C})$sl(4,C)

Abstract

The Euclidean group E(3) is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. It is the noncompact, semidirect product group \documentclass[12pt]{minimal}\begin{document}$E(3) \cong SO(3) \ltimes \mathbb {R}^{3}$\end{document}E(3)≅SO(3)⋉R3. The Euclidean algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) is the complexification of the Lie algebra of E(3). We classify the embeddings of the Euclidean algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) into the simple Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C) and, as an application of this classification, discuss the restriction to various embeddings of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) of certain irreducible representations of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C). In particular, we consider which of these restrictions are \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3)-indecomposable.