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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
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