A classification of the embeddings of the Diamond Lie algebra into $\mathfrak {sl}(3,\mathbb {C})$sl(3,C) and $\mathfrak {sp}(4,\mathbb {C})$sp(4,C), and restrictions of irreducible modules

Abstract

The Diamond Lie algebra $\mathfrak {D}$D is a four-dimensional, solvable Lie algebra. It is a central extension of the Poincaré Lie algebra in two dimensions. We classify the embeddings of $\mathfrak {D}$D into each of the classical Lie algebras of rank 2, namely, $\mathfrak {sl}(3,\mathbb {C})$sl(3,C) and $\mathfrak {sp}(4,\mathbb {C})$sp(4,C), up to conjugation by elements of $SL(3,\mathbb {C})$SL(3,C) or $Sp(4,\mathbb {C})$Sp(4,C), respectively. We then show that the irreducible representations of $\mathfrak {sl}(3,\mathbb {C})$sl(3,C) or $\mathfrak {sp}(4,\mathbb {C})$sp(4,C) remain indecomposable upon restriction to $\mathfrak {D}$D under any embedding.