Nonassociative rings and algebras Semisimple Lie algebras Concept
Root system
Summary
A root system is a finite set of vectors in a Euclidean space that is preserved by the reflections in the hyperplanes orthogonal to them. In addition, an integrality condition restricts the angles and the lengths. Root systems arise from semisimple Lie algebras over and classify them. They also govern reflection groups, algebraic groups and many finite configurations in geometry.
Where this leads
23 pages build on this one.
- Albert algebra
- Exceptional Lie algebra E6
- Zelmanov's classification of prime Jordan algebras
- Simple roots
- Exceptional Lie algebra E8
- Exceptional Lie algebra E7
- Serre's theorem on semisimple Lie algebras
- Jordan–von Neumann–Wigner classification
- Dynkin diagram
- Freudenthal magic square
- Root system of a semisimple Lie algebra
- Exceptional Lie algebra G2
- Weyl chamber
- Kostant partition function
- Cartan matrix
- Exceptional Lie algebra F4
- Coroot
- Kac–Moody algebra
- Weyl group
- Chevalley basis
- Classification of simple Lie algebras
- Classification of simple Lie superalgebras
- Root pairing
Further reading
About this page
| Tag | AVVV93CASN |
|---|---|
| URI | m2://AVVV93CASN |
| Kind | concept |
| Writing | W0 stub |
| MSC | 17B22, 20F55 |
| Same as | |
| Changed | 3 November 2026 |
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Used by
- Exceptional Lie algebra E6
- Simple roots
- Exceptional Lie algebra E8
- Exceptional Lie algebra E7
- Serre's theorem on semisimple Lie algebras
- Dynkin diagram
- Root system of a semisimple Lie algebra
- Exceptional Lie algebra G2
- Weyl chamber
- Kostant partition function
- Cartan matrix
- Exceptional Lie algebra F4
- Coroot
- Weyl group
- Chevalley basis
- Root pairing