Root system

See also Simple Lie group.

A root system is a finite set Φ of non-zero vectors (roots) spanning a finite-dimensional Euclidean space V which satisfy the following properties:

1. The only scalar multiples of a root α in V which belong to &Phi are α itself and −α.
2. For every root α in V, the set Φ is symmetric under reflection through the hyperplane of vectors perpendicular to α
3. If α and β are vectors in Φ, the projection of 2β onto the line through α is an integer multiple of α

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 (E₁,Φ₁) and (E₂,Φ₂) are considered to be the same if there is an invertible linear transformation E₁→E₂ 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.

Table of contents

1 Classification 1.1 An 1.2 Bn 1.3 Cn 1.4 Dn 1.5 En 1.6 F4 1.7 G2 2 Root systems and Lie theory

Classification

It is not too difficult to classify the root systems of rank 2:

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.

In general, irreducible root systems are specified by a family (indicated by a letter A to G) and the rank (indicated by a subscript). There are four infinite families and five exceptional cases:

• A_n (n≥1)
• B_n (n≥2)
• C_n (n≥3)
• D_n (n≥4)
• E₆
• E₇
• E₈
• F₄
• G₂

To prove this classification theorem, one uses the angles between pairs of roots to encode the root system in a much simpler combinatorial object, the Dynkin diagram. The Dynkin diagrams can then be classified according to the scheme given above.

The actual root systems can be realized as follows:

A_n

Let V be the subspace of Rⁿ⁺¹ for which the coordinates sum to 0, and let Φ be the set of vectors in V of length √2 and with integer coordinates in Rⁿ⁺¹. Such a vector must have all but two coordinates equal to 0, one coordinate equal to 1, and one equal to -1, so there are n²+n roots in all.

B_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of length 1 or √2. The total number of roots is 2n².

C_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of √2 together with all vectors of the form 2λ, where λ is an integer vector of length 1. The total number of roots is 2n². The total number of roots is 2n².

D_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of length √2. The total number of roots is 2n².

E_n

For V₈, let V=R⁸, and let E₈ denote the set of vectors α of length √2 such that the coordinates of 2α are all integers and are either all even or all odd. Then E₇ can be constructed as the intersection of E₈ with the hyperplane of vectors perpendicular to a fixed root α in E₈, and E₆ can be constructed as the intersection of E₈ with two such hyperplanes corresponding to roots α and β which are neither orthogonal to one another nor scalar multiples of one another. The root systems E₆, E₇, and E₈ have 72, 126, and 240 roots respectively.

F₄

For F₄, let V=R⁴, and let Φ denote the set of vectors α of length 1 or √2 such that the coordinates of 2α are all integers and are either all even or all odd. There are 48 roots in this system.

G₂

There are 12 roots in G₂, which form the vertices of a hexagram. See the picture above.

Root systems and Lie theory

Irreducible root systems classify a number of related objects in Lie theory, notably:

• Simple complex Lie algebras
• Simple complex Lie groups
• Simply connected complex Lie groups which are simple modulo their centers
• Simple compact Lie groups

In each case, the roots are non-zero weightss of the adjoint representation.

See also Weyl group, Coxeter group, Cartan matrix, Coxeter matrix

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A root system can also be said to describe a plant's roots and associated systems