Root system

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.^[1]

This article is about root systems in mathematics. For plant root systems, see Root.

History[edit]

The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem^[10]).^[11] He used them in his attempt to classify all simple Lie algebras over the field of complex numbers. Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F₄. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.^[12]

Killing investigated the structure of a Lie algebra $L$ , by considering what is now called a Cartan subalgebra ${\mathfrak {h}}$ . Then he studied the roots of the characteristic polynomial $\det(\operatorname {ad} _{L}x-t)$ , where $x\in {\mathfrak {h}}$ . Here a root is considered as a function of ${\mathfrak {h}}$ , or indeed as an element of the dual vector space ${\mathfrak {h}}^{*}$ . This set of roots form a root system inside ${\mathfrak {h}}^{*}$ , as defined above, where the inner product is the Killing form.^[11]

Angle of 90 degrees; in that case, the length ratio is unrestricted.

Angle of 60 or 120 degrees, with a length ratio of 1.

Angle of 45 or 135 degrees, with a length ratio of ${\sqrt {2}}$ .

Angle of 30 or 150 degrees, with a length ratio of ${\sqrt {3}}$ .

The cosine of the angle between two roots is constrained to be one-half of the square root of a positive integer. This is because $\langle \beta ,\alpha \rangle$ and $\langle \alpha ,\beta \rangle$ are both integers, by assumption, and

Since $2\cos(\theta )\in [-2,2]$ , the only possible values for $\cos(\theta )$ are $0,\pm {\tfrac {1}{2}},\pm {\tfrac {\sqrt {2}}{2}},\pm {\tfrac {\sqrt {3}}{2}}$ and $\pm {\tfrac {\sqrt {4}}{2}}=\pm 1$ , corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0° or 180°. Condition 2 says that no scalar multiples of α other than 1 and −1 can be roots, so 0 or 180°, which would correspond to 2α or −2α, are out. The diagram at right shows that an angle of 60° or 120° corresponds to roots of equal length, while an angle of 45° or 135° corresponds to a length ratio of ${\sqrt {2}}$ and an angle of 30° or 150° corresponds to a length ratio of ${\sqrt {3}}$ .

In summary, here are the only possibilities for each pair of roots.^[13]

For each root $\alpha \in \Phi$ exactly one of the roots $\alpha$ , $-\alpha$ is contained in $\Phi ^{+}$ .

For any two distinct $\alpha ,\beta \in \Phi ^{+}$ such that $\alpha +\beta$ is a root, $\alpha +\beta \in \Phi ^{+}$ .

Given a root system $\Phi$ we can always choose (in many ways) a set of positive roots. This is a subset $\Phi ^{+}$ of $\Phi$ such that

If a set of positive roots $\Phi ^{+}$ is chosen, elements of $-\Phi ^{+}$ are called negative roots. A set of positive roots may be constructed by choosing a hyperplane $V$ not containing any root and setting $\Phi ^{+}$ to be all the roots lying on a fixed side of $V$ . Furthermore, every set of positive roots arises in this way.^[14]

An element of $\Phi ^{+}$ is called a simple root (also fundamental root) if it cannot be written as the sum of two elements of $\Phi ^{+}$ . (The set of simple roots is also referred to as a base for $\Phi$ .) The set $\Delta$ of simple roots is a basis of $E$ with the following additional special properties:^[15]

For each root system $\Phi$ there are many different choices of the set of positive roots—or, equivalently, of the simple roots—but any two sets of positive roots differ by the action of the Weyl group.^[16]

No edge if the vectors are orthogonal,

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.

(see the discussion above on root systems arising from semisimple Lie algebras),

simple complex Lie algebras

complex Lie groups which are simple modulo centers, and

simply connected

compact Lie groups which are simple modulo centers.

simply connected

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

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

We now give a brief indication of how irreducible root systems classify simple Lie algebras over $\mathbb {C}$ , following the arguments in Humphreys.^[24] A preliminary result says that a semisimple Lie algebra is simple if and only if the associated root system is irreducible.^[25] We thus restrict attention to irreducible root systems and simple Lie algebras.

For connections between the exceptional root systems and their Lie groups and Lie algebras see E₈, E₇, E₆, F₄, and G₂.

The E₈ root system is any set of vectors in R⁸ that is to the following set:
$D_{8}\cup \left\{{\frac {1}{2}}\left(\sum _{i=1}^{8}\varepsilon _{i}\mathbf {e} _{i}\right):\varepsilon _{i}=\pm 1,\,\varepsilon _{1}\cdots \varepsilon _{8}=+1\right\}.$

congruent

ADE classification

Affine root system

Coxeter–Dynkin diagram

Coxeter group

Coxeter matrix

Dynkin diagram

root datum

Semisimple Lie algebra

Weights in the representation theory of semisimple Lie algebras

Root system of a semi-simple Lie algebra

Weyl group

(1983), Lectures on Lie groups, University of Chicago Press, ISBN 0-226-00530-5

Adams, J.F.

(2002), Lie groups and Lie algebras, Chapters 4–6 (translated from the 1968 French original by Andrew Pressley), Elements of Mathematics, Springer-Verlag, ISBN 3-540-42650-7. The classic reference for root systems.

Bourbaki, Nicolas

(1998). Elements of the History of Mathematics. Springer. ISBN 3540647678.

Bourbaki, Nicolas

Coleman, A.J. (Summer 1989), "The greatest mathematical paper of all time", The Mathematical Intelligencer, 11 (3): 29–38, :10.1007/bf03025189, S2CID 35487310

doi

Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, 978-3319134666

ISBN

(1972). Introduction to Lie algebras and Representation Theory. Springer. ISBN 0387900535.

Humphreys, James

Humphreys, James (1992). Reflection Groups and Coxeter Groups. Cambridge University Press. 0521436133.

ISBN

Killing, Wilhelm

(1990). Infinite-Dimensional Lie Algebras (3rd ed.). Cambridge University Press. ISBN 978-0-521-46693-6.

Kac, Victor G.

Springer, T.A. (1998). Linear Algebraic Groups (2nd ed.). Birkhäuser. 0817640215.

ISBN

Dynkin, E.B. (1947). . Uspekhi Mat. Nauk. 2 (in Russian). 4 (20): 59–127. MR 0027752.

Root system

History[edit]

simple complex Lie algebras

simply connected

simply connected

congruent

ADE classification

Affine root system

Coxeter–Dynkin diagram

Coxeter group

Coxeter matrix

Dynkin diagram

root datum

Semisimple Lie algebra

Weights in the representation theory of semisimple Lie algebras

Root system of a semi-simple Lie algebra

Weyl group

Adams, J.F.

Bourbaki, Nicolas

Bourbaki, Nicolas

doi

ISBN

Humphreys, James

ISBN

Killing, Wilhelm

Kac, Victor G.

ISBN

"The structure of semi-simple algebras"