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Concept
Root system
Summary
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory.
As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors span the whole space. If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, to another root. Moreover, the root to which it is sent equals α + nβ, where n is an integer (in this case, n equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one is known as A2.
Let E be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by . A root system in E is a finite set of non-zero vectors (called roots) that satisfy the following conditions:
The roots span E.
The only scalar multiples of a root that belong to are itself and .
For every root , the set is closed under reflection through the hyperplane perpendicular to .
(Integrality) If and are roots in , then the projection of onto the line through is an integer or half-integer multiple of .
An equivalent way of writing conditions 3 and 4 is as follows:
For any two roots , the set contains the element
For any two roots , the number is an integer.
Official source
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