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Classification of invariant cones in Lie algebras

Joachim Hilgert, Karl H. Hofmann

1988
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Bulletin of the American Mathematical Society
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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
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... 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

doi:10.1090/s0273-0979-1988-15692-3
fatcat:3acljwsdxbfy3fiesg6dtopwgu