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On closed subsets of root systems. (English) Zbl 0808.17015
Summary: Let $$R$$ be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space $$V$$. A subset $$P\subset R$$ is closed if $$\alpha, \beta \in P$$ and $$\alpha+ \beta\in R$$ imply that $$\alpha+ \beta\in P$$. In this paper we classify, up to conjugacy by the Weyl group $$W$$ of $$R$$, all closed sets $$P\subset R$$ such that $$R\setminus P$$ is also closed. We also show that if $$\theta: R\to R'$$ is a bijection between two root systems such that both $$\theta$$ and $$\theta^{-1}$$ preserve closed sets, and if $$R$$ has at most one irreducible component of type $$A_ 1$$, then $$\theta$$ is an isomorphism of root systems.

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
##### Keywords:
root system; Weyl group; closed sets
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