Let be a Euclidean space, be the dot product, and denote the reflection in
the hyperplane
by

where

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

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

2. If , the reflection leaves invariant, and

3. If ,
then .

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
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
Chamber,

Weyl's Denominator Formula,

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
Web Resource. https://mathworld.wolfram.com/RootSystem.html

## Subject classifications