Lie algebras with complex structures having nilpotent eigenspaces

Abstract

Let (g, [•, •]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of gC isomorphic to the algebra (g, [*]J) with bracket [X * Y]J = 2 ([X, Y] - [JX, JY]). We consider here the case where these subalgebras are nilpotent and prove that the original (g, [•, •]) Lie algebra must be solvable. We consider also the 6-dimensional case and determine explicitly the possible nilpotent Lie algebras (g, [*]J). Finally we produce several examples illustrating different situations, in particular we show that for each given s there exists g with complex structure J such that (g, [*]J) is s-step nilpotent. Similar examples of hypercomplex structures are also built.