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Invariant cones in Lie algebras. (English) Zbl 0648.22005

A cone in a finite dimensional real Lie algebra is a closed convex subset stable under the scalar multiplication by nonnegative real numbers; it is, therefore, an additive semigroup. A cone W in a Lie algebra \({\mathfrak g}\) is called invariant if \(e^{ad x}(W)=W\) for all \(x\in {\mathfrak g}\). The theory of invariant cones in Lie algebras is an important part of the Lie theory of semigroups. Invariant cones arise as the set of (“one- sided”) tangent vectors at 0 of the pull-back \(\exp^{-1}(S)\) of any semigroup S in a Lie group G with the exponential function exp: \({\mathfrak g}\to G\) provided this semigroup is invariant under inner automorphisms. Every domain of positivity for a partial order on G satisfying both left and right monotonicity laws is such a semigroup. In the paper the authors completely describe invariant cones in Lie algebras.
Reviewer: A.K.Guts

MSC:

22E15 General properties and structure of real Lie groups
17B20 Simple, semisimple, reductive (super)algebras
22A15 Structure of topological semigroups
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