Root System

Let E be a Euclidean space, (beta,alpha) be the dot product, and denote the reflection in the hyperplane P_alpha={beta in E|(beta,alpha)=0} by

where

Then a subset R of the Euclidean space E is called a root system in E if:

1. R is finite, spans E, and does not contain 0,

2. If alpha in R, the reflection sigma_alpha leaves R invariant, and

3. If alpha,beta in R, then <beta,alpha> in Z.

The Lie algebra roots of a semisimple Lie algebra are a root system, in a real subspace of the dual vector space to the Cartan subalgebra. In this case, the reflections W_alpha generate the Weyl group, which is the symmetry group of the root system.

See also

Cartan Matrix, Lie Algebra, Lie Algebra Root, Lie Algebra Weight, Macdonald's Constant-Term Conjecture, Reduced Root System, Semisimple Lie Algebra, Weyl Group

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Root System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RootSystem.html

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