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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
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s 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 groups and Lie algebras, especially the classification and representation theory of
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra i ...
s. Since Lie groups (and some analogues such as
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
s) 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 diagram In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
s, occurs in parts of mathematics with no overt connection to Lie theory (such as
singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
). 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 ...
, R2, 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 ester ...
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 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 ''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 (\cdot,\cdot). A root system \Phi 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 ester ...
''E''. # The only scalar multiples of a root \alpha\in\Phi that belong to \Phi are \alpha itself and -\alpha. # For every root \alpha\in\Phi, the set \Phi is closed under
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
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 \alpha. # (Integrality) If \alpha and \beta are roots in \Phi, then the projection of \beta onto the line through \alpha is an ''integer or half-integer'' multiple of \alpha. An equivalent way of writing conditions 3 and 4 is as follows: #
  • For any two roots \alpha,\beta\in\Phi, the set \Phi contains the element \sigma_\alpha(\beta):=\beta-2\frac\alpha.
  • # For any two roots \alpha,\beta\in\Phi, the number \langle \beta, \alpha \rangle := 2 \frac 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 \langle \cdot, \cdot \rangle \colon \Phi \times \Phi \to \mathbb 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, Φ1) and (''E''2, Φ2) are called isomorphic if there is an invertible linear transformation ''E''1 → ''E''2 which sends Φ1 to Φ2 such that for each pair of roots, the number \langle x, y \rangle is preserved. The of a root system Φ is the Z-submodule of ''E'' generated by Φ. It is a
    lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
    in ''E''.


    Weyl group

    The
    group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
    of
    isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
    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 A_2 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 A_1.


    Rank two examples

    In rank 2 there are four possibilities, corresponding to \sigma_\alpha(\beta) = \beta + n\alpha, where n = 0, 1, 2, 3. The figure at right shows these possibilities, but with some redundancies: A_1\times A_1 is isomorphic to D_2 and B_2 is isomorphic to C_2. Note that a root system is not determined by the lattice that it generates: A_1 \times A_1 and B_2 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 A_2 and G_2 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 \mathfrak is a complex
    semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra i ...
    and \mathfrak is a
    Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
    , we can construct a root system as follows. We say that \alpha\in\mathfrak^* is a root of \mathfrak relative to \mathfrak if \alpha\neq 0 and there exists some X\neq 0\in\mathfrak such that ,X\alpha(H)X for all H\in\mathfrak. One can show that there is an inner product for which the set of roots forms a root system. The root system of \mathfrak is a fundamental tool for analyzing the structure of \mathfrak 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 s ...
    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 F4. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic. Killing investigated the structure of a Lie algebra L, by considering what is now called a
    Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
    \mathfrak. 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 ...
    \det (\operatorname_L x - t), where x \in \mathfrak. Here a ''root'' is considered as a function of \mathfrak, or indeed as an element of the dual vector space \mathfrak^*. This set of roots form a root system inside \mathfrak^*, 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 \langle \beta, \alpha \rangle and \langle \alpha, \beta \rangle are both integers, by assumption, and \begin \langle \beta, \alpha \rangle \langle \alpha, \beta \rangle &= 2 \frac \cdot 2 \frac \\ &= 4 \frac \\ &= 4 \cos^2(\theta) = (2\cos(\theta))^2 \in \mathbb. \end Since 2\cos(\theta) \in 2,2/math>, the only possible values for \cos(\theta) are 0, \pm \tfrac, \pm\tfrac, \pm\tfrac and \pm\tfrac = \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 2''α'' or −2''α'', 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 and an angle of 30° or 150° corresponds to a length ratio of \sqrt. In summary, here are the only possibilities for each pair of roots. *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.


    Positive roots and simple roots

    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 * 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^+. 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. 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: *Every root \alpha\in\Phi is a linear combination of elements of \Delta with ''integer'' coefficients. *For each \alpha\in\Phi, the coefficients in the previous point are either all non-negative or all non-positive. 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.


    Dual root system, coroots, and integral elements


    The dual root system

    If Φ is a root system in ''E'', the coroot α of a root α is defined by \alpha^\vee= \, \alpha. The set of coroots also forms a root system Φ in ''E'', called the dual root system (or sometimes ''inverse root system''). By definition, α∨ ∨ = α, so that Φ is the dual root system of Φ. The lattice in ''E'' spanned by Φ is called the ''coroot lattice''. Both Φ and Φ have the same Weyl group ''W'' and, for ''s'' in ''W'', (s\alpha)^\vee= s(\alpha^\vee). If Δ is a set of simple roots for Φ, then Δ is a set of simple roots for Φ. In the classification described below, the root systems of type A_n and D_n along with the exceptional root systems E_6,E_7,E_8,F_4,G_2 are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the B_n and C_n root systems are dual to one another, but not isomorphic (except when n=2).


    Integral elements

    A vector \lambda in ''E'' is called integral if its inner product with each coroot is an integer: 2\frac\in\mathbb Z,\quad\alpha\in\Phi. Since the set of \alpha^\vee with \alpha\in\Delta forms a base for the dual root system, to verify that \lambda is integral, it suffices to check the above condition for \alpha\in\Delta. The set of integral elements is called the weight lattice associated to the given root system. This term comes from the
    representation theory of semisimple Lie algebras In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is a ...
    , where the integral elements form the possible weights of finite-dimensional representations. The definition of a root system guarantees that the roots themselves are integral elements. Thus, every integer linear combination of roots is also integral. In most cases, however, there will be integral elements that are not integer combinations of roots. That is to say, in general the weight lattice does not coincide with the root lattice.


    Classification of root systems by Dynkin diagrams

    A root system is irreducible if it cannot be partitioned into the union of two proper subsets \Phi=\Phi_1\cup\Phi_2, such that (\alpha,\beta)=0 for all \alpha\in\Phi_1 and \beta\in\Phi_2 . Irreducible root systems correspond to certain
    graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
    , the
    Dynkin diagram In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
    s named after
    Eugene Dynkin Eugene Borisovich Dynkin (russian: link=no, Евгений Борисович Дынкин; 11 May 1924 – 14 November 2014) was a USSR, Soviet and American mathematician. He made contributions to the fields of probability and algebra, especial ...
    . The classification of these graphs is a simple matter of
    combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
    , and induces a classification of irreducible root systems.


    Constructing the Dynkin diagram

    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 the roots in Δ. Edges are drawn between vertices as follows, according to the angles. (Note that the angle between simple roots is always at least 90 degrees.) *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. The term "directed edge" means that double and triple edges are marked with an arrow pointing toward the shorter vector. (Thinking of the arrow as a "greater than" sign makes it clear which way the arrow is supposed to point.) Note that by the elementary properties of roots noted above, the rules for creating the Dynkin diagram can also be described as follows. No edge if the roots are orthogonal; for nonorthogonal roots, a single, double, or triple edge according to whether the length ratio of the longer to shorter is 1, \sqrt 2, \sqrt 3. In the case of the G_2 root system for example, there are two simple roots at an angle of 150 degrees (with a length ratio of \sqrt 3). Thus, the Dynkin diagram has two vertices joined by a triple edge, with an arrow pointing from the vertex associated to the longer root to the other vertex. (In this case, the arrow is a bit redundant, since the diagram is equivalent whichever way the arrow goes.)


    Classifying root systems

    Although a given root system has more than one possible set of simple roots, 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 ...
    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. A root systems is irreducible if and only if its Dynkin diagrams is connected. The possible connected diagrams are as indicated in the figure. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system). If \Phi is a root system, the Dynkin diagram for the dual root system \Phi^\vee is obtained from the Dynkin diagram of \Phi by keeping all the same vertices and edges, but reversing the directions of all arrows. Thus, we can see from their Dynkin diagrams that B_n and C_n are dual to each other.


    Weyl chambers and the Weyl group

    If \Phi\subset E is a root system, we may consider the hyperplane perpendicular to each root \alpha. Recall that \sigma_\alpha denotes the reflection about the hyperplane and that 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 ...
    is the group of transformations of E generated by all the \sigma_\alpha's. The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber. If we have fixed a particular set Δ of simple roots, we may define the fundamental Weyl chamber associated to Δ as the set of points v\in E such that (\alpha,v)>0 for all \alpha\in\Delta. Since the reflections \sigma_\alpha,\,\alpha\in\Phi preserve \Phi, they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers. The figure illustrates the case of the A_2 root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base. A basic general theorem about Weyl chambers is this: :Theorem: The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl chambers. In the A_2 case, for example, the Weyl group has six elements and there are six Weyl chambers. A related result is this one: :Theorem: Fix a Weyl chamber C. Then for all v\in E, the Weyl-orbit of v contains exactly one point in the closure \bar C of C.


    Root systems and Lie theory

    Irreducible root systems classify a number of related objects in Lie theory, notably the following: * simple complex Lie algebras (see the discussion above on root systems arising from semisimple Lie algebras), *
    simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
    complex Lie groups which are simple modulo centers, and *
    simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
    compact Lie groups which are simple modulo centers. In each case, the roots are non-zero
    weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
    s of the
    adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
    . We now give a brief indication of how irreducible root systems classify simple Lie algebras over \mathbb C, following the arguments in Humphreys. A preliminary result says that a
    semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra i ...
    is simple if and only if the associated root system is irreducible. We thus restrict attention to irreducible root systems and simple Lie algebras. *First, we must establish that for each simple algebra \mathfrak g there is only one root system. This assertion follows from the result that the Cartan subalgebra of \mathfrak g is unique up to automorphism, from which it follows that any two Cartan subalgebras give isomorphic root systems. *Next, we need to show that for each irreducible root system, there can be at most one Lie algebra, that is, that the root system determines the Lie algebra up to isomorphism. *Finally, we must show that for each irreducible root system, there is an associated simple Lie algebra. This claim is obvious for the root systems of type A, B, C, and D, for which the associated Lie algebras are the
    classical Lie algebras The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A_n , B_n , C_n and D_n , where for \mathfrak(n) the General linear group, general linear Lie algebra and I_n the n \times n ...
    . It is then possible to analyze the exceptional algebras in a case-by-case fashion. Alternatively, one can develop a systematic procedure for building a Lie algebra from a root system, using Serre's relations. For connections between the exceptional root systems and their Lie groups and Lie algebras see E8, E7, E6, F4, and G2.


    Properties of the 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). The subscript indicates the rank of the root system. In an irreducible root system there can be at most two values for the length (''α'', ''α'')1/2, corresponding to short and long roots. If all roots have the same length they are taken to be long by definition and the root system is said to be simply laced; this occurs in the cases A, D and E. Any two roots of the same length lie in the same orbit of the Weyl group. In the non-simply laced cases B, C, G and F, the root lattice is spanned by the short roots and the long roots span a sublattice, invariant under the Weyl group, equal to ''r''2/2 times the coroot lattice, where ''r'' is the length of a long root. In the adjacent table, , Φ<, denotes the number of short roots, ''I'' denotes the index in the root lattice of the sublattice generated by long roots, ''D'' denotes the determinant of the
    Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Ki ...
    , and , ''W'', denotes the order of 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 ...
    .


    Explicit construction of the irreducible root systems


    A''n''

    Let ''E'' be the subspace of R''n''+1 for which the coordinates sum to 0, and let Φ be the set of vectors in ''E'' of length and which are ''integer vectors,'' i.e. have integer coordinates in R''n''+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 ''n''2 + ''n'' roots in all. One choice of simple roots expressed in the
    standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...
    is: α''i'' = e''i'' − e''i''+1, for 1 ≤ ''i'' ≤ n. The
    reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
    ''σ''''i'' 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 α''i'' is the same as
    permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
    of the adjacent ''i''-th and (''i'' + 1)-th
    coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
    . Such transpositions generate the full
    permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...
    . For adjacent simple roots, ''σ''''i''''i''+1) = α''i''+1 + α''i'' = ''σ''''i''+1''i'') = α''i'' + α''i''+1, that is, reflection is equivalent to adding a multiple of 1; but reflection of a simple root perpendicular to a nonadjacent simple root leaves it unchanged, differing by a multiple of 0. The ''A''''n'' root lattice – that is, the lattice generated by the ''A''''n'' roots – is most easily described as the set of integer vectors in R''n''+1 whose components sum to zero. The A2 root lattice is the
    vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...
    of the
    triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
    . The A3 root lattice is known to crystallographers as the
    face-centered cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of ...
    (or cubic close packed) lattice. It is the vertex arrangement of the
    tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names i ...
    . The A3 root system (as well as the other rank-three root systems) may be modeled in the Zometool
    Construction set A construction set is a set of standardized pieces that allow for the construction of a variety of different models. The pieces avoid the lead-time of manufacturing custom pieces, and of requiring special training or design time to constr ...
    . In general, the A''n'' root lattice is the vertex arrangement of the ''n''-dimensional
    simplectic honeycomb In geometry, the simplectic honeycomb (or -simplex honeycomb) is a dimensional infinite series of honeycombs, based on the _n affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of nodes with one nod ...
    .


    B''n''

    Let ''E'' = R''n'', and let Φ consist of all integer vectors in ''E'' of length 1 or . The total number of roots is 2''n''2. One choice of simple roots is: α''i'' = e''i'' – e''i''+1, for 1 ≤ ''i'' ≤ ''n'' – 1 (the above choice of simple roots for A''n''−1), and the shorter root α''n'' = e''n''. The reflection ''σ''''n'' through the hyperplane perpendicular to the short root α''n'' is of course simply negation of the ''n''th coordinate. For the long simple root α''n''−1, σ''n''−1''n'') = α''n'' + α''n''−1, but for reflection perpendicular to the short root, ''σ''''n''''n''−1) = α''n''−1 + 2α''n'', a difference by a multiple of 2 instead of 1. The ''B''''n'' root lattice – that is, the lattice generated by the ''B''''n'' roots – consists of all integer vectors. ''B''1 is isomorphic to A1 via scaling by , and is therefore not a distinct root system.


    C''n''

    Let ''E'' = R''n'', and let Φ consist of all integer vectors in ''E'' of length together with all vectors of the form 2''λ'', where ''λ'' is an integer vector of length 1. The total number of roots is 2''n''2. One choice of simple roots is: α''i'' = e''i'' − e''i''+1, for 1 ≤ ''i'' ≤ ''n'' − 1 (the above choice of simple roots for A''n''−1), and the longer root α''n'' = 2e''n''. The reflection ''σ''''n''''n''−1) = α''n''−1 + α''n'', but ''σ''''n''−1''n'') = α''n'' + 2α''n''−1. The ''C''''n'' root lattice – that is, the lattice generated by the ''C''''n'' roots – consists of all integer vectors whose components sum to an even integer. C2 is isomorphic to B2 via scaling by and a 45 degree rotation, and is therefore not a distinct root system.


    D''n''

    Let ''E'' = R''n'', and let Φ consist of all integer vectors in ''E'' of length . The total number of roots is 2''n''(''n'' − 1). One choice of simple roots is: α''i'' = e''i'' − e''i''+1, for 1 ≤ ''i'' ≤ ''n'' − 1 (the above choice of simple roots for A''n''−1) plus α''n'' = e''n'' + e''n''−1. Reflection through the hyperplane perpendicular to α''n'' is the same as transposing and negating the adjacent ''n''-th and (''n'' − 1)-th coordinates. Any simple root and its reflection perpendicular to another simple root differ by a multiple of 0 or 1 of the second root, not by any greater multiple. The ''D''''n'' root lattice – that is, the lattice generated by the ''D''''n'' roots – consists of all integer vectors whose components sum to an even integer. This is the same as the ''C''''n'' root lattice. The ''D''''n'' roots are expressed as the vertices of a rectified ''n''-
    orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
    , Coxeter-Dynkin diagram: .... The 2''n''(''n''−1) vertices exist in the middle of the edges of the ''n''-orthoplex. D3 coincides with A3, and is therefore not a distinct root system. The 12 D3 root vectors are expressed as the vertices of , a lower symmetry construction of the
    cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
    . D4 has additional symmetry called
    triality In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin( ...
    . The 24 D4 root vectors are expressed as the vertices of , a lower symmetry construction of the
    24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
    .


    E6, E7, E8

    *The ''E''8 root system is any set of vectors in R8 that is
    congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
    to the following set: D_8 \cup \left\. The root system has 240 roots. The set just listed is the set of vectors of length in the E8 root lattice, also known simply as the
    E8 lattice In mathematics, the E lattice is a special lattice in R. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E root system. The normIn t ...
    or Γ8. This is the set of points in R8 such that: #all the coordinates are
    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 ...
    s or all the coordinates are
    half-integer In mathematics, a half-integer is a number of the form :n + \tfrac, where n is an whole number. For example, :, , , 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
    s (a mixture of integers and half-integers is not allowed), and #the sum of the eight coordinates is an
    even integer In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ ...
    . Thus, E_8 = \left\ * The root system E7 is the set of vectors in E8 that are perpendicular to a fixed root in E8. The root system E7 has 126 roots. * The root system E6 is not the set of vectors in E7 that are perpendicular to a fixed root in E7, indeed, one obtains D6 that way. However, E6 is the subsystem of E8 perpendicular to two suitably chosen roots of E8. The root system E6 has 72 roots. An alternative description of the E8 lattice which is sometimes convenient is as the set Γ'8 of all points in R8 such that *all the coordinates are integers and the sum of the coordinates is even, or *all the coordinates are half-integers and the sum of the coordinates is odd. The lattices Γ8 and Γ'8 are
    isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
    ; one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ8 is sometimes called the ''even coordinate system'' for E8 while the lattice Γ'8 is called the ''odd coordinate system''. One choice of simple roots for E8 in the even coordinate system with rows ordered by node order in the alternate (non-canonical) Dynkin diagrams (above) is: :α''i'' = e''i'' − e''i''+1, for 1 ≤ ''i'' ≤ 6, and :α7 = e7 + e6 (the above choice of simple roots for D7) along with \mathbf\alpha_8 = \mathbf\beta_0 = -\frac \left( \sum_^8e_i\right) = (-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2). One choice of simple roots for E8 in the odd coordinate system with rows ordered by node order in alternate (non-canonical) Dynkin diagrams (above) is: :α''i'' = e''i'' − e''i''+1, for 1 ≤ ''i'' ≤ 7 (the above choice of simple roots for A7) along with :α8 = β5, where :βj = \frac \left(- \sum_^j e_i + \sum_^8 e_i\right). (Using β3 would give an isomorphic result. Using β1,7 or β2,6 would simply give A8 or D8. As for β4, its coordinates sum to 0, and the same is true for α1...7, so they span only the 7-dimensional subspace for which the coordinates sum to 0; in fact −2β4 has coordinates (1,2,3,4,3,2,1) in the basis (α''i'').) Since perpendicularity to α1 means that the first two coordinates are equal, E7 is then the subset of E8 where the first two coordinates are equal, and similarly E6 is the subset of E8 where the first three coordinates are equal. This facilitates explicit definitions of E7 and E6 as: :E''7'' = , :E''6'' = Note that deleting α1 and then α2 gives sets of simple roots for E7 and E6. However, these sets of simple roots are in different E7 and E6 subspaces of E8 than the ones written above, since they are not orthogonal to α1 or α2.


    F4

    For F4, let ''E'' = R4, and let Φ denote the set of vectors α of length 1 or such that the coordinates of 2α are all integers and are either all even or all odd. There are 48 roots in this system. One choice of simple roots is: the choice of simple roots given above for B3, plus \boldsymbol\alpha_4 = -\frac \sum_^4 e_i. The F4 root lattice – that is, the lattice generated by the F4 root system – is the set of points in R4 such that either all the coordinates are
    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 ...
    s or all the coordinates are
    half-integer In mathematics, a half-integer is a number of the form :n + \tfrac, where n is an whole number. For example, :, , , 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
    s (a mixture of integers and half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions.


    ''G''2

    The root system G2 has 12 roots, which form the vertices of a
    hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...
    . See the picture above. One choice of simple roots is: (α1, β = α2 − α1) where α''i'' = e''i'' − e''i''+1 for ''i'' = 1, 2 is the above choice of simple roots for ''A''2. The ''G''2 root lattice – that is, the lattice generated by the ''G''2 roots – is the same as the ''A''2 root lattice.


    The root poset

    The set of positive roots is naturally ordered by saying that \alpha \leq \beta if and only if \beta-\alpha is a nonnegative linear combination of simple roots. This
    poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
    is graded by \deg\left(\sum_ \lambda_\alpha \alpha\right) = \sum_\lambda_\alpha, and has many remarkable combinatorial properties, one of them being that one can determine the degrees of the fundamental invariants of the corresponding Weyl group from this poset. The Hasse graph is a visualization of the ordering of the root poset.


    See also

    *
    ADE classification In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
    *
    Affine root system In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple ''p''-adic algebraic groups, and correspond to f ...
    *
    Coxeter–Dynkin diagram In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describe ...
    *
    Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
    *
    Coxeter matrix In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
    *
    Dynkin diagram In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
    *
    root datum In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, publish ...
    *
    Semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra i ...
    * Weights in the representation theory of semisimple Lie algebras *
    Root system of a semi-simple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra ...
    *
    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 ...


    Notes


    References

    * *. The classic reference for root systems. * * * * * * ** ** ** * *


    Further reading

    *


    External links

    {{Authority control Euclidean geometry Lie groups Lie algebras