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Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
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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 …