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root system (Definition)

A root system is a key notion in the classification and the representation theory of reflection groups and of semi-simple Lie algebras. Let $ E$ be a Euclidean vector space with inner product $ (\cdot,\cdot)$. A root system is a finite spanning set $ R\subset E$ such that for every $ u\in R$, the orthogonal reflection

$\displaystyle v\mapsto v-2\frac{(u,v)}{(u,u)} u,\quad v\in E$
preserves $ R$.

A root system is called crystallographic if $ 2\frac{(u,v)}{(u,u)}$ is an integer for all $ u,v\in R$.

A root system is called reduced if for all $ u\in R$, we have $ ku\in R$ for $ k=\pm 1$ only.

We call a root system indecomposable if there is no proper decomposition $ R=R'\cup R''$ such that every vector in $ R'$ is orthogonal to every vector in $ R''$.



"root system" is owned by rmilson. [ full author list (2) | owner history (1) ]
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See Also: simple and semi-simple Lie algebras, Lie algebra

Also defines:  reduced root system, root, root space, root decomposition, indecomposable, reduced, crystallographic

Attachments:
classification of indecomposable root systems (Result) by rmilson
root system underlying a semi-simple Lie algebra (Result) by rmilson
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Cross-references: vector, decomposition, integer, preserves, orthogonal, spanning set, finite, inner product, Euclidean vector space, semi-simple Lie algebras, groups, reflection, theory, representation
There are 94 references to this entry.

This is version 10 of root system, born on 2002-12-04, modified 2006-12-12.
Object id is 3645, canonical name is RootSystem.
Accessed 20242 times total.

Classification:
AMS MSC: 17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive )
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