The Poincaré algebra in rank 3 simple Lie algebras

摘要

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