Semi-simple real subalgebras of non-compact semi-simple real lie algebras. I

Abstract

A detailed and systematic study is made of the problem of embedding a semi-simple real Lie algebra L ′ in a non-compact semi-simple real Lie algebra L . The analysis is based on the work of Cartan and Gantmacher in which non-compact real forms are generated from compact real forms using the involutive automorphisms of the compact real forms. A simple necessary condition and a convenient necessary and sufficient condition for embedding L ′ in L are derived. The situation in which L ′ and L are generated by inner involutive automorphisms is then examined in detail for the case in the compact form corresponding to L is a classical Lie algebra, and a procedure is formulated for constructing for all the possible embeddings of L ′ in L . This procedure is illustrated by considering a number of examples, including those for which the corresponding complex Lie algebra embeddings are A l ⊂ D l +1, C l ⊂ A 2 l-1 , D l ⊂ B l , and B l ⊂ D l+1 , for which some new results are obtained. The situation in which L ′ or L are generated by outer involutive automorphisms will be considered in other papers of this series.