Classification of invariant cones in Lie algebras

Joachim Hilgert, Karl H. Hofmann
1988 Bulletin of the American Mathematical Society
All Lie algebras in the following are finite dimensional real Lie algebras. A cone in a finite dimensional real vector space is a closed convex subset stable under the scalar multiplication by the set R + of nonnegative real numbers; it is, therefore additively closed and may contain vector subspaces. A cone W in a Lie algebra g is called invariant if (1) e * dx {W)=W for all z eg. We shall describe invariant cones in Lie algebras completely. For simple Lie algebras see [KR82, 0181, Pa84, and
more » ... 80]. Some observations are simple: If W is an invariant cone in a Lie algebra g, then the edge e = W Pi -W and the span W -W are ideals. Therefore, if one aims for a theory without restriction on the algebra g it is no serious loss of generality to assume that W is generating, that is, satisfies g = W -W. This is tantamount to saying that W has inner points. Also, the homomorphic image W/t is an invariant cone with zero edge in the algebra g/e. Therefore, nothing is lost if we assume that W is pointed, that is, has zero edge. Invariant pointed generating cones can for instance be found in sl(2,R), the oscillator algebra and compact Lie algebras with nontrivial center (see [HH85b, c, HH86a, or HHL87]). A subalgebra \) of a Lie algebra g is said to be compactly embedded if the analytic group Inn 0 f) generated by the set e ad *> in Aut g has a compact closure. Even for a compactly embedded Cartan algebra I) of a solvable algebra g, the analytic group Inn 0 f) need not be closed in Aut 0 [HH86]. An element x G g is called compact if R • x is a compactly embedded subalgebra, and the set of all compact elements of g will be denoted compg. It is true, although not entirely superficial that a superalgebra is compactly embedded if and only if it is contained in compg. l. THEOREM (THE UNIQUENESS THEOREM [HH86b]). Let W be an invariant pointed generating cone in a Lie algebra g. Then (i) 'mtW Ç compg. (ii) If H is any compactly embedded Cartan algebra, then (a) HD'mtW ^0, and (b) int W = (Inn 0 g) int" ftnW). In particular, compactly embedded Cartan algebras exist, and if f)i and fo #re compactly embedded Cartan algebras and W\ and W<2 are invariant pointed generating cones of g such that t)C\Wi = \) D W2, then W\ = VK 2 . D