The Poincaré algebra in rank 3 simple Lie algebras

Abstract

We classify embeddings of the Poincaré algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {p}(3,1)$\end{document}p(3,1) into the rank 3 simple Lie algebras. Up to inner automorphism, we show that there are exactly two embeddings of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {p}(3,1)$\end{document}p(3,1) into \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C), which are, however, related by an outer automorphism of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C). Next, we show that there is a unique embedding of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {p}(3,1)$\end{document}p(3,1) into \documentclass[12pt]{minimal}\begin{document}$\mathfrak {so}(7,\mathbb {C})$\end{document}so(7,C), up to inner automorphism, but no embeddings of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {p}(3,1)$\end{document}p(3,1) into \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sp}(6,\mathbb {C})$\end{document}sp(6,C). All embeddings are explicitly described. As an application, we show that each irreducible highest weight module of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C) (not necessarily finite-dimensional) remains indecomposable when restricted to \documentclass[12pt]{minimal}\begin{document}$\mathfrak {p}(3,1)$\end{document}p(3,1), with respect to any embedding of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {p}(3,1)$\end{document}p(3,1) into \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C).