In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a root system is a configuration of
vectors in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
satisfying certain geometrical properties. The concept is fundamental in the theory of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s and
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s, 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
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix ...
.
Definitions and examples
As a first example, consider the six vectors in 2-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, R
2, as shown in the image at the right; call them roots. These vectors
span
Span may refer to:
Science, technology and engineering
* Span (unit), the width of a human hand
* Span (engineering), a section between two intermediate supports
* Wingspan, the distance between the wingtips of a bird or aircraft
* Sorbitan es ...
the whole space. If you consider the line
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to any root, say ''β'', then the reflection of R
2 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 ''A''
2.
Definition
Let ''E'' be a finite-dimensional
Euclidean vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, 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
Span may refer to:
Science, technology and engineering
* Span (unit), the width of a human hand
* Span (engineering), a section between two intermediate supports
* Wingspan, the distance between the wingtips of a bird or aircraft
* Sorbitan es ...
''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
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
.
Some authors only include conditions 1–3 in the definition of a root system. In this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2; then they call root systems satisfying condition 2 reduced. In this article, all root systems are assumed to be reduced and crystallographic.
In view of property 3, the integrality condition is equivalent to stating that ''β'' and its reflection ''σ''
''α''(''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 ''E''.
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 ''A''
2, ''B''
2, and ''G''
2 pictured to the right, is said to be irreducible.
Two root systems (''E''
1, Φ
1) and (''E''
2, Φ
2) are called isomorphic if there is an invertible linear transformation ''E''
1 → ''E''
2 which sends Φ
1 to Φ
2 such that for each pair of roots, the number
is preserved.
The of a root system Φ is the Z-submodule of ''E'' generated by Φ. It is a
lattice in ''E''.
Weyl group
The
group of
isometries of ''E'' generated by reflections through hyperplanes associated to the roots of Φ is called the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
of Φ. As it
acts faithfully on the finite set Φ, the Weyl group is always finite. The reflection planes are the hyperplanes perpendicular to the roots, indicated for
by dashed lines in the figure below. The Weyl group is the symmetry group of an equilateral triangle, which has six elements. In this case, the Weyl group is not the full symmetry group of the root system (e.g., a 30-degree rotation (order 6) is a symmetry of the root system but not an element of the Weyl group).
Rank one example
There is only one root system of rank 1, consisting of two nonzero vectors
. This root system is called
.
Rank two examples
In rank 2 there are four possibilities, corresponding to
, where
. The figure at right shows these possibilities, but with some redundancies:
is isomorphic to
and
is isomorphic to
.
Note that a root system is not determined by the lattice that it generates:
and
both generate a
square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their ...
while
and
generate a
hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° ...
, only two of the five possible types of
lattices in two dimensions.
Whenever Φ is a root system in ''E'', and ''S'' is a
subspace of ''E'' spanned by Ψ = Φ ∩ ''S'', then Ψ is a root system in ''S''. Thus, the exhaustive list of four root systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank. In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.
Root systems arising from semisimple Lie algebras
If
is a complex
semisimple Lie algebra and
is a
Cartan subalgebra, we can construct a root system as follows. We say that
is a root of
relative to
if
and there exists some
such that
for all
. One can show that there is an inner product for which the set of roots forms a root system. The root system of
is a fundamental tool for analyzing the structure of
and classifying its representations. (See the section below on Root systems and Lie theory.)
History
The concept of a root system was originally introduced by
Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
Life
Killing studied at the University of Mü ...
around 1889 (in German, ''Wurzelsystem'').
He used them in his attempt to classify all
simple Lie algebra
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan.
A direct sum of si ...
s over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. 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 F
4. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.
Killing investigated the structure of a Lie algebra
, by considering what is now called a
Cartan subalgebra . Then he studied the roots of the
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
, where
. Here a ''root'' is considered as a function of
, or indeed as an element of the dual vector space
. This set of roots form a root system inside
, as defined above, where the inner product is the
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
.
[
]
Elementary consequences of the root system axioms
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 and are both integers, by assumption, and
Since