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The Weyl group of a root system is a subgroup generated by reflections through the hyperplanes orthogonal to the roots.
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Why Weyl group associated to Cartan matrix which defines positive definite bilinear form is ...
In the book by V.Kac "Infinite dimensional Lie algebras" in proposition 4.9 it is argued that Weyl group constructed by Cartan matrix which defines positive-definite metric on Cartan subalgebra is fin …