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positive root (Definition)

If $ R\subset E$ is a root system, with $ E$ an Euclidean vector space, then a subset $ R^+\subset R$ is called a set of positive roots if there is a vector $ v\in E$ such that $ (\alpha ,v)>0$ if $ \alpha\in R^+$, and $ (\alpha ,v)<0$ if $ \alpha\in R\backslash R^+$. Roots which are not positive are called negative. Since $ -\alpha$ is negative if and only if $ \alpha$ is positive, exactly half the roots must be positive.



"positive root" is owned by mathwizard. [ full author list (2) | owner history (1) ]
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Also defines:  negative root
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Cross-references: negative, positive, vector, subset, Euclidean vector space, root system
There are 13 references to this entry.

This is version 5 of positive root, born on 2002-12-04, modified 2004-04-13.
Object id is 3656, canonical name is PositiveRoot.
Accessed 4432 times total.

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