Conjugacy of Cartan Subalgebras in Solvable Leibniz Algebras and Real Leibniz Algebras

Abstract

ABSTRACT. We extend conjugacy results from Lie algebras to their Leibniz algebra generalizations. The proofs in the Lie case depend on anti-commutativity. Thus it is necessary to find other paths in the Leibniz case. Some of these results involve Cartan subalgebras. Our results can be used to extend other results on Cartan subalgebras. We show an example here and others will be shown in future work.Keywords: Leibniz algebras, Conjugacy, Cartan subalgebras, maximal subalgebras MSC 2010: 17A32 A self-centralizing minimal ideal in a solvable Lie algebra is complemented and all complements are conjugate [1]. Such an algebra is called primitive. If the minimal ideal is not self-centralizing, then there is a bijection which assigns complements to their intersection with the centralizer [7]. Such an intersection is the core of the complement. Towers extended this result to show that maximal subalgebras are conjugate exactly when their cores coincide, with the additional condition at characteristic p that L 2 has nilpotency class less than p. Under this same condition, Cartan subalgebras are conjugate. We show these results hold for Leibniz algebras. The antisymmetry is used in the Lie algebra case, and so we amend the arguments in the Leibniz algebra case. We also show consequences for the conjugacy of Cartan subalgebras in solvable Lie algebras. We also consider conjugacy of Cartan subalgebras for real Leibniz algebras, obtaining an extension of a result of Barnes

Classification of five-dimensional symmetric Leibniz algebras

“…In recent years, the theory of Leibniz algebras has been intensively studied. During the last 30 years the theory of Leibniz algebras has been actively studied and many results on Lie algebras have been extended to Leibniz algebras (see for example [5,6,16,17,23,28,29]).…”