Duality theories for metabelian Lie algebras

Abstract

This paper is concerned with duality theories for metabelian (2-step nilpotent) Lie algebras. A duality theory associates to each metabelian Lie algebra N with cod N 2 = g {N^2} = g , another such algebra N D {N_D} satisfying ( N D ) D ≅ N , N 1 ≅ N 2 {({N_D})_D} \cong N,{N_1} \cong {N_2} if and only if ( N 1 ) D ≅ ( N 2 ) D {({N_1})_D} \cong {({N_2})_D} , and if dim N = g + p \dim \,N = g + p then dim N D = g + ( 2 g ) − p \dim \,{N_D} = g + (_2^g) - p . The obvious benefit of such a theory lies in its reduction of the classification problem.