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In Mathematics , a root system is a configuration of Vector s in a Euclidean Space satisfying certain geometrical properties. The concept is fundamental in Lie Group theory. Since Lie groups (and some analogues such as Algebraic Group s) have come to be used in most 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 groups (such as Singularity Theory ). DEFINITIONS Let ''V'' be a finite-dimensional Euclidean Space , with the standard Euclidean Inner Product denoted by (·,·). A root system in ''V'' is a finite set Φ of non-zero vectors (called '''roots''') that satisfy the following properties: # The roots Span ''V'' # The only scalar multiples of a root α ∈ Φ that belong to Φ are α itself and −α. # For every root α ∈ Φ, the set Φ is closed under Reflection through the Hyperplane perpendicular to α. That is, for any two roots α and β, #: # (''Integrality condition'') If α and β are roots in Φ, then the projection of β onto the line through α is a half-integral multiple of α. That is, #: In view of property 3, the integrality condition is equivalent to stating that β and its reflection σα(β) differ by an integer multiple of α. Note that the operator : defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument. The rank of a root system Φ is the dimension of ''V''. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A2, B2, and G2 pictured below, is said to be irreducible. Two irreducible root systems (''E''1,Φ1) and (''E''2,Φ2) are considered to be the same if there is an invertible linear transformation ''E''1→''E''2 which preserves distance up to a scale factor and which sends Φ1 to Φ2. The Group of Isometries of ''V'' generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl Group of Φ. As it Acts Faithfully on the finite set Φ, the Weyl group is always finite. RANK 1 AND RANK 2 EXAMPLES There is only one root system of rank 1, consisting of two nonzero vectors {α, −α}. This root system is called A1. In rank 2 there are four possibilities: Whenever Φ is a root system in ''V'' and ''W'' is a Subspace of ''V'' spanned by Ψ=Φ∩''W'', then Ψ is a root system in ''W''. Thus, our exhaustive list of root systems of rank 2 shows the geometric possibilities for any two roots in a root system. In particular, two such roots meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees. POSITIVE ROOTS AND SIMPLE ROOTS Given a root system Φ we can always choose (in many ways) a set of positive roots. This is a subset of Φ such that
If a set of positive roots is chosen, elements of () are called negative roots. The choice of is equivalent to the choice of simple roots. The set of simple roots is a subset Δ of Φ which is a basis of ''V'' with the special property that every vector in Φ when written in the basis Δ has either all coefficients ≥0 or else all ≤0. It can be shown that for each choice of positive roots there exists a unique set of simple roots so that the positive roots are exactly those roots that can be expressed as a combination of simple roots with non-negative coefficients. CLASSIFICATION OF ROOT SYSTEMS BY DYNKIN DIAGRAMS Irreducible root systems Correspond to certain Graphs , the Dynkin diagrams named for Eugene Dynkin . The classification of these graphs is a simple matter of Combinatorics , and induces a classification of irreducible root systems. Given a root system, select a set Δ of Simple Roots as in the preceding section. The vertices of the associated Dynkin diagram correspond to vectors in Δ. An edge is drawn between each non-orthogonal pair of vectors; it is an undirected single edge if they make an angle of 120 degrees, a directed double edge if they make an angle of 135 degrees, and a directed triple edge if they make an angle of 150 degrees. In addition, double and triple edges are marked with an angle sign pointing toward the shorter vector. Although a given root system has more than one possible set of simple roots, the Weyl Group acts transitively on such choices. Consequently, the Dynkin diagram is independent of the choice of simple roots; it is determined by the root system itself. Conversely, given two root systems with the same Dynkin diagram, one can match up roots, starting with the roots in the base, and show that the systems are in fact the same. Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams. The problem of classifying irreducible root systems reduces to the problem of classifying connected Dynkin diagrams. Dynkin diagrams encode the inner product on ''E'' in terms of the basis Δ, and the condition that this inner product must be Positive Definite turns out to be all that is needed to get the desired classification. The actual connected diagrams are as follows. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system). PROPERTIES OF IRREDUCIBLE ROOT SYSTEMS Irreducible root systems are named according to their corresponding connected Dynkin diagrams. There are four infinite families (A''n'', B''n'', C''n'', and D''n'', called the classical root systems) and five exceptional cases (the '''exceptional root systems'''). In every case, the subscript indicates the rank of the root system. The following table lists some of their other properties. |
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