Structure of sympathetic 3-Lie algebras

In this paper, we will introduce the concept of sympathetic [Formula: see text]-Lie algebras and show that some classical properties of semi-simple [Formula: see text]-Lie algebras are still valid for sympathetic [Formula: see text]-Lie algebras. We prove that every perfect [Formula: see text]-Lie algebra [Formula: see text] contains a greatest special sympathetic ideal [Formula: see text], and there exists a solvable ideal of [Formula: see text] denoted by [Formula: see text] which is the greatest among the solvable ideals [Formula: see text] of [Formula: see text] for which [Formula: see text]. And we show that there exists a sympathetic subalgebra [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is a sympathetic [Formula: see text]-Lie algebra if and only if [Formula: see text]. Moreover, we also study the ideals [Formula: see text] of a [Formula: see text]-Lie algebra [Formula: see text] such that [Formula: see text] is a sympathetic [Formula: see text]-Lie algebra and investigate some properties about them.

Proof of the radical conjecture for homogeneous Kähler manifolds

In 1967 Gindikin and Vinberg stated the Fundamental Conjecture for homogeneous Kähler manifolds. It (roughly) states that every homogeneous Kähler manifold is a fiber space over a bounded homogeneous domain for which the fibers are a product of a flat with a simply connected compact homogeneous Kähler manifold. This conjecture has been proven in a number of cases (see [6] for a recent survey). In particular, it holds if the homogeneous Kähler manifold admits a reductive or an arbitrary solvable transitive group of automorphisms [5]. It is thus tempting to think about the general case. It is natural to expect that lack of knowledge about the radical of a transitive group G of automorphisms of a homogeneous Kähler manifold M is the main obstruction to a proof of the Fundamental Conjecture for M. Thus it is of importance to consider the Kähler algebra generated by the radical of the Lie algebra of G. Computations in this context suggest that one rather considers Kähler algebras generated by an arbitrary solvable ideal.