Classification of Invariant Cones in Lie Algebras

We shall describe invariant cones in Lie algebras completely. For simple Lie algebras see [KR82, 0181, Pa84, and Vi80]. 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 Inn0 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 Inn0 f) need not be closed in Aut0 [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.