Orthogonal lie algebras with cone potential

Abstract

We say that a Lie algebra g is orthogonal if it admits a non-degenerate invariant symmetric bilinear form. In this paper we give a complete decription of all orthogonal Lie algebras with cone potential, a property defined in terms of the root decomposition with respect to a compactly embedded Cartan subalgebra. The class of these Lie algebras arises naturally in the study of convexity properties of Hamiltonian torus actions which are related to decompositions of groups similar to Iwasawa decompositions G = KAN of real semisimple Lie groups. In [HiNe94] we have generalized Kostant’s nonlinear convexity theorem for the Iwasawa decomposition for a complex Lie group to the class of complex groups whose Lie algebras have a real form which is an orthogonal Lie algebra with cone potential. We even obtained a generalization to the class of real semisimple Lie algebras having a solvable minimal parabolic subalgebra. This work generalized results of Lu and Ratiu ([LR91]) and aims at a unified symplectic framework for the convexity theorem in [Ne94] and the linear convexity theorems for coadjoint orbits of convexity type in [HNP93]. The motivation for this paper is to clarify the scope of the convexity theorems proved in [HiNe94]. Our description gives a rather clear picture of the situation. Every orthognal Lie algebra with cone potential is the ideal direct sum of a solvable Lie algebra and a semisimple Lie algebra of the same type. A semisimple Lie algebras has cone potential if and only if has a compactly embedded Cartan subalgebra. The orthogonal solvable Lie algebras with cone potential also decompose into certain indecomposable pieces which can be described by a root decomposition.