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Root system

A root system is a finite collection of nonzero vectors, known as roots, in a finite-dimensional Euclidean space that spans the space, includes negatives of all roots, is invariant under reflections across hyperplanes perpendicular to any root, and satisfies an integrality condition ensuring that the inner products between roots yield integer values when normalized.^[1] These structures are "reduced" in the sense that the only scalar multiples of a root within the system are itself and its negative.^[1] Root systems form the foundational geometric framework for understanding the structure of semisimple Lie algebras over the complex numbers, where the roots correspond to the eigenvalues of the adjoint representation of the Cartan subalgebra.^[2] The key properties of a root system include the generation of a finite Weyl group by the reflections associated with each root, which acts faithfully on the space and permutes the roots.^[1] Positive roots can be defined relative to a choice of simple roots forming a basis, and the entire system is generated from these via the Weyl group action.^[2] Root systems are classified up to isomorphism into irreducible types, corresponding to connected Dynkin diagrams: four infinite families (A_n for n ≥ 1, B_n for n ≥ 2, C_n for n ≥ 3, D_n for n ≥ 4) and five exceptional cases (E_6, E_7, E_8, F_4, G_2).^[1] This classification, established in the mid-20th century, mirrors the classification of simple Lie algebras and provides a combinatorial tool via Dynkin diagrams to encode the angles and relative lengths between simple roots.^[2] Beyond Lie theory, root systems appear in the study of reductive algebraic groups and their representations, as well as in finite reflection groups and Coxeter systems.^[1] The "crystallographic" condition—requiring integer structure constants—ensures compatibility with the integer lattices arising in these contexts, distinguishing them from more general finite reflection arrangements.^[2] Historically, root systems were formalized in the 1950s by Claude Chevalley and others building on Élie Cartan's work on Lie algebras, with axiomatic definitions refined in the 1960s by Alexander Grothendieck to emphasize connections to algebraic groups.^[1]

Core Definitions and Structures

Definition of a Root System

In mathematics, particularly in the study of Lie algebras and reflection groups, a root system is a fundamental structure consisting of a finite set \Phi of nonzero vectors in a finite-dimensional real Euclidean space E equipped with a positive definite inner product (\cdot, \cdot). The set \Phi must span E, and it satisfies two key axioms: first, it is invariant under reflections across the hyperplanes perpendicular to its elements, meaning that for every \alpha \in \Phi and \beta \in \Phi, the reflection s_\alpha(\beta) = \beta - 2 \frac{(\beta, \alpha)}{(\alpha, \alpha)} \alpha also lies in \Phi; second, it obeys an integrality condition, where $2 \frac{(\beta, \alpha)}{(\alpha, \alpha)} \in \mathbb{Z} for all \alpha, \beta \in \Phi.^[3]^[4] This axiomatic framework captures the geometric properties of roots arising from semisimple Lie algebras, where the reflections generate a finite Weyl group acting on E. Root systems are classified into types based on additional constraints: reduced root systems, the most common type, require that no vector in \Phi is a scalar multiple of another except for \pm 1 times itself, ensuring distinct lengths up to sign; non-reduced root systems relax this by permitting other scalar multiples, such as twice a short root alongside long and short roots, as seen in certain extended systems like those associated to affine Lie algebras; crystallographic (or crystalline) root systems emphasize the integrality axiom, allowing the roots to generate a full-rank lattice in E over the integers, which is crucial for connections to integer matrices and Weyl group representations.^[3]^[4]^[5] The rank of a root system \Phi is defined as the dimension of the span of \Phi in E, which equals \dim E since \Phi spans E. A root system is irreducible if it cannot be decomposed as an orthogonal direct sum of two nonempty proper subsystems, a property that corresponds to indecomposable Dynkin diagrams in the classification of finite root systems.^[3]^[5]

Weyl Group Action

The Weyl group W(\Phi) of a root system \Phi in the real Euclidean space E (with positive definite inner product (\cdot, \cdot)) is defined as the subgroup of the orthogonal group O(E) generated by the reflections s_\alpha for all roots \alpha \in \Phi. The reflection across the hyperplane perpendicular to \alpha is given by the formula

s_\alpha(v) = v - 2 \frac{(\alpha, v)}{(\alpha, \alpha)} \alpha

for all v \in E. Each such reflection is an orthogonal transformation, preserving the inner product, and lies in O(E).^[6] The Weyl group preserves the root system, meaning w(\Phi) = \Phi for every w \in W(\Phi). To see this, note that for any root \beta \in \Phi, the reflection s_\alpha(\beta) = \beta - k \alpha where k = 2 (\alpha, \beta)/(\alpha, \alpha) is an integer by the axioms of the root system; thus s_\alpha(\beta) \in \Phi unless k = 0, in which case it fixes \beta. Since the reflections map \Phi to itself and W(\Phi) is generated by these reflections, the entire group acts on \Phi. Moreover, in an irreducible root system, W(\Phi) acts transitively on the set of roots of any given length. The proof follows from the connectedness of the Coxeter graph associated to the simple roots and the fact that reflections generate transformations between roots at the same angle to a fixed one, covering all possibilities within each length class.^[6]^[7] A root system \Phi is crystallographic if, for all \alpha, \beta \in \Phi, the Cartan integer $2 (\beta, \alpha)/(\alpha, \alpha) is an integer. This integrality condition ensures that the bilinear form restricts to an integral structure on the root lattice, and that the Weyl group W(\Phi) is a finite Coxeter group. Specifically, W(\Phi) admits a presentation as a Coxeter group with generators corresponding to reflections across simple root hyperplanes and relations determined by the angles between them, guaranteeing finiteness via the positive definiteness of the form and the discrete nature of the reflections.^[8] The order of the Weyl group admits an explicit formula as the product of the degrees of its fundamental invariants under the action on the polynomial ring of E. These degrees are even integers greater than or equal to 2, specific to each root system type (e.g., 2, 3, ..., r+1 for type A_r), and their product yields the group order; this arises from the structure theorem for invariants of finite reflection groups. For instance, in rank 2, the order is 8 for type B_2 (degrees 2 and 4) and 12 for type G_2 (degrees 2 and 6).^[9]

Positive Roots and Simple Roots

In a root system \Phi spanning a Euclidean space E, a positive subsystem \Phi^+ is a proper subset satisfying \Phi = \Phi^+ \sqcup (-\Phi^+) and such that if \alpha, \beta \in \Phi^+ with \alpha + \beta \in \Phi, then \alpha + \beta \in \Phi^+.^[10] This additivity condition ensures that \Phi^+ consists precisely of the roots that lie on one side of a hyperplane arrangement defined by the root hyperplanes, corresponding to a choice of Weyl chamber. Such subsystems exist for any root system and are unique up to the action of the Weyl group W, which acts simply transitively on the set of positive subsystems.^[10] Given a positive subsystem \Phi^+, the simple roots \Delta \subset \Phi^+ form a base: they are a linearly independent subset that spans E over \mathbb{R}, and every \alpha \in \Phi^+ admits a unique expression \alpha = \sum_{\delta \in \Delta} k_\delta \delta with coefficients k_\delta \in \mathbb{Z}_{\geq 0}, while every \beta \in -\Phi^+ has all negative integer coefficients.^[10] Moreover, for distinct \alpha, \beta \in \Delta, the inner product satisfies (\alpha, \beta) \leq 0, and \alpha - \beta \notin \Phi. The simple roots are precisely the indecomposable elements of \Phi^+, meaning those that cannot be expressed as a sum of two nonzero elements of \Phi^+.^[10] Every root system admits at least one such base, and any two bases \Delta, \Delta' are conjugate under an element of the Weyl group W.^[10] In the representation-theoretic context of semisimple Lie algebras \mathfrak{g}, a choice of positive subsystem corresponds to a Borel subalgebra \mathfrak{b} \subset \mathfrak{g} containing a Cartan subalgebra \mathfrak{h}, where the unipotent radical of \mathfrak{b} is spanned by the root spaces \mathfrak{g}_\alpha for \alpha \in \Phi^+; the simple roots then index the minimal parabolic subalgebras containing \mathfrak{b}.^[11] The Cartan matrix associated to an ordered base \Delta = \{\alpha_1, \dots, \alpha_l\} (where l = \dim E) is the integer matrix A = (a_{ij}) with entries

a_{ij} = \frac{2(\alpha_i, \alpha_j)}{(\alpha_i, \alpha_i)},

so that the diagonal entries are a_{ii} = 2 and the off-diagonal entries satisfy a_{ij} \leq 0 for i \neq j.^[10] These off-diagonal values are constrained to \{0, -1, -2, -3\}, and the matrix is independent of the specific ordering up to simultaneous permutation of rows and columns by the action of the Weyl group on the base.^[10] The Cartan matrix uniquely determines the root system up to isomorphism.^[10]

Low-Rank Examples and Motivations

Rank-One Root Systems

The rank-one root system represents the simplest non-trivial configuration in the theory of root systems, residing in a one-dimensional Euclidean space. It serves as a foundational example that illustrates core properties such as reflections and positivity without the complexity of higher dimensions. In the reduced case, which is the standard focus for finite irreducible root systems, the set of roots is \Phi = \{\alpha, -\alpha\} for some nonzero vector \alpha \in \mathbb{R} with positive squared length (\alpha, \alpha) > 0.^[1]^[5] This setup satisfies the axioms of a root system, where the reflection s_\alpha across the hyperplane perpendicular to \alpha maps \alpha to -\alpha and preserves the set \Phi.^[1] The Weyl group of this root system is the finite group generated by s_\alpha, which is isomorphic to \mathbb{Z}/2\mathbb{Z} and acts by interchanging the two roots.^[12]^[1] Selecting a positive subsystem yields the single positive root \{\alpha\}, with \alpha itself serving as the unique simple root, forming a basis for the root lattice.^[12]^[5] The associated Cartan matrix is the trivial $1 \times 1 matrix

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, reflecting the integer structure constant \langle \alpha, \alpha^\vee \rangle = 2, where \alpha^\vee = 2\alpha / (\alpha, \alpha) is the coroot.^[1]^[12] A non-reduced variant extends the reduced system by including multiples, such as \Phi = \{\pm \alpha, \pm 2\alpha\} (often denoted BC_1), where \alpha has unit length for normalization; however, the reduced case remains the primary example in classical Lie theory due to its irreducibility and correspondence to the Lie algebra \mathfrak{sl}_2(\mathbb{C}).^[13] Geometrically, the roots lie symmetrically on the real line, with the Weyl group action manifesting as point reflection through the origin, underscoring the system's inherent bilateral symmetry.^[1]^[5]

Rank-Two Root Systems

The irreducible root systems of rank two consist of four distinct types: the reduced systems A_2, B_2, and G_2, along with the non-reduced system BC_2. These systems are realized in a two-dimensional Euclidean space and are classified up to isomorphism based on the possible angles between roots and the ratios of root lengths, which determine their geometric configurations.^[14]^[11]^[15] The A_2 root system features six roots of equal length forming a hexagonal lattice in the plane, with angles between adjacent roots of $60^\circ or $120^\circ. An explicit realization embeds it in \mathbb{R}^3 with roots \{\pm(e_1 - e_2), \pm(e_2 - e_3), \pm(e_3 - e_1)\}, where these vectors span the plane orthogonal to e_1 + e_2 + e_3 and all have squared length 2. The Weyl group is the dihedral group of order 6, isomorphic to the symmetric group S_3, acting as rotations and reflections preserving the equilateral triangular arrangement of simple roots. The Cartan matrix is

\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix},

reflecting the equal-length roots and $120^\circ angle between simple roots.^[11]^[16]^[14] In contrast, the B_2 root system includes eight roots with two distinct lengths: four short roots of squared length 1 and four long roots of squared length 2, yielding a length ratio of \sqrt{2}. A standard realization in \mathbb{R}^2 uses roots \{\pm e_1, \pm e_2, \pm(e_1 + e_2), \pm(e_1 - e_2)\}, where angles between roots are multiples of $45^\circ, including $90^\circ and $135^\circ between simple roots. The Weyl group is the dihedral group of order 8, corresponding to the symmetries of a square. Its Cartan matrix, distinguishing short and long roots, is

\begin{pmatrix} 2 & -1 \\ -2 & 2 \end{pmatrix}.

^[11]^[16]^[14] The G_2 root system has twelve roots with two lengths: six short roots of squared length 2 and six long roots of squared length 6, for a ratio of \sqrt{3}. Realized in \mathbb{R}^3 spanning a plane, the roots are \{\pm(e_1 - e_2), \pm(e_1 - e_3), \pm(e_2 - e_3), \pm(2e_1 - e_2 - e_3), \pm(2e_2 - e_1 - e_3), \pm(2e_3 - e_1 - e_2)\}, with angles that are multiples of $30^\circ, such as $30^\circ and $150^\circ between simple roots. The Weyl group is the dihedral group of order 12, reflecting the higher symmetry from the length disparity. The Cartan matrix is

\begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix},

capturing the tripled off-diagonal entry due to the short root's coroot.^[11]^[16]^[14] Finally, the non-reduced BC_2 root system extends B_2 by including roots of three distinct lengths—short (squared length 1), medium (\sqrt{2}), and long (2)—with twelve roots total in \mathbb{R}^2: \{\pm e_1, \pm e_2, \pm 2e_1, \pm 2e_2, \pm(e_1 \pm e_2)\}. This introduces multiples of short roots, violating the reduced condition, while maintaining angles as multiples of $45^\circ, including $90^\circ and $135^\circ. The Weyl group remains the dihedral group of order 8, and it shares the Cartan matrix of B_2:

\begin{pmatrix} 2 & -1 \\ -2 & 2 \end{pmatrix}.

^[17]^[18]^[14]

Origins in Semisimple Lie Algebras

In the theory of semisimple Lie algebras over the complex numbers, root systems arise naturally from the structure of these algebras and their Cartan subalgebras. Consider a semisimple Lie algebra \mathfrak{g} equipped with a Cartan subalgebra \mathfrak{h}, which is a maximal abelian subalgebra consisting of semisimple elements. The roots are defined as the elements of the set \Phi = \{\alpha \in \mathfrak{h}^* \mid \mathfrak{g}_\alpha \neq 0\}, where \mathfrak{g}_\alpha = \{x \in \mathfrak{g} \mid [h, x] = \alpha(h) x \text{ for all } h \in \mathfrak{h}\}.^[19] This leads to the root space decomposition of \mathfrak{g}: \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha. For semisimple Lie algebras over \mathbb{C}, each root space \mathfrak{g}_\alpha is one-dimensional.^[19] The Killing form B(X, Y) = \operatorname{tr}(\operatorname{ad}_X \operatorname{ad}_Y) on \mathfrak{g} is nondegenerate and symmetric, restricting to a nondegenerate bilinear form on \mathfrak{h}. This induces a Euclidean inner product on the real span of \Phi in \mathfrak{h}^*, endowing \Phi with the structure of a root system.^[19] A concrete example occurs in the special linear Lie algebra \mathfrak{sl}(n+1, \mathbb{C}), whose root system is of type A_n. Here, \mathfrak{h} consists of trace-zero diagonal matrices, and the roots are \alpha_{ij} = e_i - e_j for i \neq j, where \{e_1, \dots, e_{n+1}\} is the standard basis of \mathbb{C}^{n+1}.^[19]

Basic Properties and Dualities

Axiomatic Consequences

The reflections associated to roots in a root system are linear transformations of the underlying Euclidean space E. Specifically, for roots \alpha, \beta \in \Phi, the reflection s_\alpha(\beta) = \beta - 2 \proj_\alpha(\beta), where \proj_\alpha(\beta) = \frac{(\beta, \alpha)}{(\alpha, \alpha)} \alpha. This formula defines an orthogonal reflection over the hyperplane perpendicular to \alpha, preserving the inner product and thus acting as a linear isometry on E.^[3] A fundamental consequence of the axioms is the root string property. For fixed \alpha, \beta \in \Phi with \beta \not\propto \alpha, the set of integers k such that \beta + k \alpha \in \Phi forms a consecutive string \beta - p \alpha, \dots, \beta + q \alpha with p, q \geq 0 integers satisfying p + q \leq 3, so the string has length at most 4. Moreover, p - q = \frac{2(\beta, \alpha)}{(\alpha, \alpha)}, which is an integer by the axioms. This follows from the integrality of Cartan integers and the fact that reflections s_\alpha map roots to roots while reversing the string; applying s_\alpha repeatedly shows that gaps in the string would contradict the preservation of \Phi.^[3] The string property implies that the possible values of the Cartan integer \frac{2(\beta, \alpha)}{(\alpha, \alpha)} lie in \{-3, -2, -1, 0, 1, 2, 3\} for distinct non-proportional roots, restricting the possible angles between any two roots to a finite set (such as $0, \pi/6, \pi/4, \pi/3, \pi/2, 2\pi/3, 3\pi/4, 5\pi/6, \pi). Consequently, the directions of roots on the unit sphere are confined to finitely many possibilities, so even without assuming finiteness in the axioms, the set \Phi must be finite for reduced root systems (those with no roots that are nonzero integer multiples of others beyond \pm 1).^[20] From these angle restrictions and the string property, the squared lengths (\alpha, \alpha) for \alpha \in \Phi take values in a finite set. Specifically, the ratios of squared lengths between any two roots are rational (determined by the Cartan integers), ensuring commensurability. In particular, irreducible root systems admit at most two distinct root lengths.^[3] Additionally, the set \Phi spans E over \mathbb{Q}, meaning the \mathbb{Q}-vector space generated by \Phi has dimension equal to \dim E; this follows from the linear independence over \mathbb{R} of a basis extracted from the reflections and the rationality of inner products via Cartan integers.^[21]

Dual Root System and Coroots

Given a root system \Phi in a finite-dimensional Euclidean space E equipped with a positive definite inner product (\cdot, \cdot), the dual space E^* is canonically identified with E via this inner product. Under this identification, the dual root system is defined as \Phi^\vee = \{\alpha^\vee = \frac{2\alpha}{(\alpha, \alpha)} \mid \alpha \in \Phi\} \subset E.^[10] The elements \alpha^\vee are called coroots.^[22] The set \Phi^\vee itself forms a root system in E, sharing the same Weyl group as \Phi.^[10] This dual structure preserves key geometric properties while potentially interchanging the roles of long and short roots in non-simply-laced cases.^[23] Coroots pair integrally with roots through the expression (\beta, \alpha^\vee) = \frac{2(\beta, \alpha)}{(\alpha, \alpha)} \in \mathbb{Z} for \beta \in \Phi, which encodes the Cartan integers central to the root system's structure.^[10] In the realization of root systems arising from semisimple Lie algebras \mathfrak{g} with Cartan subalgebra \mathfrak{h}, each coroot \alpha^\vee lies in \mathfrak{h} (via identification using the Killing form), and the adjoint representation satisfies [\alpha^\vee, X] = (\beta, \alpha^\vee) X for X \in \mathfrak{g}_\beta, the root space corresponding to root \beta.^[22] This action highlights the coroots' role in governing the linear transformations induced by the Lie algebra on its root spaces.^[23]

Integral Elements and Lattices

In the context of a root system \Phi in a finite-dimensional Euclidean space E equipped with a positive definite inner product (\cdot, \cdot), the root lattice Q(\Phi) is defined as the \mathbb{Z}-span of \Phi, i.e., Q(\Phi) = \mathbb{Z}\Phi = \sum_{\alpha \in \Phi} \mathbb{Z} \alpha.^[24] This forms a full-rank lattice in E, meaning its rank equals \dim E = l, the rank of the root system, and it is a discrete subgroup that spans E over \mathbb{R}.^[25] Similarly, the coroot lattice Q(\Phi^\vee) is the \mathbb{Z}-span of the coroots \Phi^\vee = \{\alpha^\vee \mid \alpha \in \Phi\}, where \alpha^\vee = 2\alpha / (\alpha, \alpha), and it also has full rank l in the dual space E^*.^[8] These lattices capture the integral structure underlying the root system, with the pairing (\lambda, \alpha^\vee) inducing connections between roots and coroots. The weight lattice P(\Phi) consists of all elements \lambda \in E such that the pairing (\lambda, \alpha^\vee) \in \mathbb{Z} for every coroot \alpha^\vee \in \Phi^\vee.^[24] This defines a full-rank lattice containing the root lattice as a sublattice, Q(\Phi) \subseteq P(\Phi), with finite index [P(\Phi) : Q(\Phi)] that depends on the type of the root system and reflects its representation-theoretic properties.^[26] In particular, P(\Phi) serves as the ambient lattice for integral weights in the associated Lie theory, where elements of P(\Phi) pair integrally with all coroots. A basis for P(\Phi) is provided by the fundamental weights \{\omega_i\}_{i=1}^l, defined relative to a base of simple coroots \{\alpha_i^\vee\}_{i=1}^l by the conditions (\omega_i, \alpha_j^\vee) = \delta_{ij} for i,j = 1, \dots, l.^[8] These \omega_i form the dual basis to the simple coroots and generate P(\Phi) over \mathbb{Z}, ensuring that every integral weight is an integer linear combination of the fundamental weights. In terms of dualities, the weight lattice P(\Phi) is the dual lattice to the coroot lattice, P(\Phi) = Q(\Phi^\vee)^*, where the dual is taken with respect to the inner product pairing; this identification holds in standard normalizations and underscores the symmetry between roots and coroots.^[26]

Classification Framework

Dynkin Diagrams Construction

Dynkin diagrams provide a graphical encoding of the structure of a root system through its simple roots, facilitating the classification of irreducible root systems associated with semisimple Lie algebras.^[27] Each diagram consists of nodes and edges that capture the inner product relations among the simple roots \Pi = \{\alpha_1, \dots, \alpha_r\}, where r is the rank of the root system.^[1] The construction begins by representing each simple root \alpha_i \in \Pi as a node in the diagram. Nodes i and j are connected by an edge if and only if the inner product (\alpha_i, \alpha_j) < 0, indicating that the roots are adjacent in the root poset and not orthogonal.^[27] The nature of the edge—its multiplicity and any direction—is determined by the Cartan integers a_{ij} = 2 (\alpha_i, \alpha_j) / (\alpha_i, \alpha_i) and a_{ji} = 2 (\alpha_j, \alpha_i) / (\alpha_j, \alpha_j), which are off-diagonal entries of the Cartan matrix. For simple root systems, these integers satisfy a_{ij} \in \{0, -1, -2, -3\} when i \neq j, with the product a_{ij} a_{ji} dictating the bond type: 1 for a single bond, 2 for a double bond, and 3 for a triple bond.^[1] When all simple roots have equal length, the diagram is undirected, with bond multiplicities reflecting symmetric relations: a single bond for |a_{ij}| = |a_{ji}| = 1, double for 2, and triple for 3. In cases of unequal root lengths, an arrow is added to the bond, pointing from the longer root to the shorter one; the multiplicity is based on the larger absolute value of a_{ij} and a_{ji}. For instance, if |a_{ij}| = 1 and |a_{ji}| = 2, a double bond with an arrow pointing to the shorter root (node j) is drawn. This convention ensures the diagram visually encodes the asymmetry in lengths.^[27] The full algorithm for construction proceeds as follows: first, identify the simple roots from the positive roots via a choice of Weyl chamber or linear functional that separates them. Then, compute the pairwise Cartan integers using the invariant bilinear form on the Euclidean space containing the root system. Finally, draw the nodes linearly (for classical types) or in the standard configuration (for exceptional types), adding edges according to the rules above. The resulting diagram uniquely determines the Cartan matrix up to permutation of nodes and duality, where duality swaps the roles of roots and coroots, interchanging arrow directions and distinguishing systems like B_n and C_n.^[1] Representative examples illustrate the construction. For the A_n series (n \geq 1), the simple roots form a chain where consecutive pairs have a_{ij} = a_{ji} = -1, yielding a linear diagram of n nodes connected by single bonds, such as \circ - \circ - \cdots - \circ for A_3.^[27] In the B_n series (n \geq 2), the diagram is a chain of n-1 single bonds followed by a double bond with an arrow pointing to the final (shorter) root, as in B_2: \circ \Rightarrow{=} \circ, where the last Cartan integers are a_{n-1,n} = -1 and a_{n,n-1} = -2.^[1] The exceptional G_2 diagram features two nodes connected by a triple bond with an arrow to the shorter root: \circ \Rightarrow{===} \circ, corresponding to a_{12} = -1 and a_{21} = -3. These examples highlight how the diagram compactly represents the off-diagonal structure of the Cartan matrix without explicit coordinates.^[27]

Irreducible Root Systems Overview

Finite irreducible root systems are classified into four infinite classical families and five exceptional cases, up to isomorphism, according to the Cartan-Killing theorem, which establishes that these are the only possibilities for reduced root systems associated to semisimple Lie algebras over the complex numbers.^[28] This classification is encoded in their Dynkin diagrams, which are graphs whose vertices correspond to a basis of simple roots and edges reflect the angles between them, with multiple edges indicating non-orthogonal pairs and arrows denoting relative lengths.^[29] The classical series consist of A_n for n ≥ 1, whose Dynkin diagram is a linear chain of n vertices connected by single edges and corresponds to the root system of the special unitary group SU(n+1); B_n for n ≥ 2, with a linear chain of n vertices where the final edge is double with an arrow pointing to the end vertex (indicating the short root there), associated to SO(2n+1); C_n for n ≥ 3, similar to B_n but with the arrow on the final double edge pointing away from the end vertex, linked to the symplectic group Sp(2n); and D_n for n ≥ 4, a linear chain of n-2 vertices followed by a fork into two additional vertices from the penultimate one, corresponding to SO(2n).^[30] The exceptional irreducible root systems are G_2, with two vertices joined by a triple edge with an arrow pointing to one vertex; F_4, a chain of four vertices with single edges for the first two, a double arrow on the third, and a single edge to the fourth; E_6, a chain of five vertices with a branch from the third to an additional vertex; E_7, a chain of six vertices with a similar branch from the third; and E_8, a chain of seven vertices with the branch from the third, forming an extended linear structure. Key invariants include the number of roots |Φ|, which for an irreducible system of rank l equals twice the number of positive roots, typically listed explicitly—for example, |A_n| = n(n+1) and |B_n| = 2n^2—reflecting the structure's scale.^[10] The order of the Weyl group |W|, the finite reflection group generated by the root system, is also characteristic: |W(A_n)| = (n+1)!, |W(B_n)| = |W(C_n)| = 2^n n!, |W(D_n)| = 2^{n-1} n!, |W(G_2)| = 12, |W(F_4)| = 1152, |W(E_6)| = 51840, |W(E_7)| = 2903040, and |W(E_8)| = 696729600.^[10] These systems are reduced, meaning no root is a scalar multiple of another except by ±1, distinguishing them from non-reduced cases like BC_n where short roots satisfy 2α being a long root; the Cartan-Killing theorem confirms no other reduced irreducible finite root systems exist beyond those listed.^[1]

Geometric and Group-Theoretic Aspects

Weyl Chambers and Group Orbits

The hyperplanes H_\alpha = \{ x \in E \mid \langle x, \alpha \rangle = 0 \} for each root \alpha \in \Phi form an arrangement that divides the finite-dimensional Euclidean space E containing the root system into open connected components, which are called the Weyl chambers.^[6] These chambers are the maximal open convex sets avoiding all root hyperplanes, and their closures include portions of the bounding hyperplanes.^[6] The number of such chambers equals the order of the Weyl group W, as W permutes them.^[6] A distinguished Weyl chamber, known as the fundamental chamber C, is defined by the inequalities \langle x, \alpha_i \rangle > 0 for all simple roots \alpha_i in a choice of positive roots \Delta^+.^[6] This choice of positive roots determines the simple roots and thus the inequalities bounding C. The Weyl group W, generated by reflections across the hyperplanes H_{\alpha_i}, acts transitively on the set of all Weyl chambers: for any two chambers C' and C'', there exists w \in W such that w(C') = C''.^[6] Moreover, this action is simply transitive, meaning the stabilizer of any chamber under W is trivial.^[6] The walls of the fundamental chamber C are the hyperplanes H_{\alpha_i} for the simple roots \alpha_i, which form its bounding facets.^[31] In the affine extension of the root system, the affine Weyl group \tilde{W} = W \ltimes Q, where Q is the root lattice, acts on E by incorporating translations, leading to an infinite arrangement of affine hyperplanes H_{\alpha, k} = \{ x \in E \mid \langle x, \alpha \rangle = k \} for \alpha \in \Phi and k \in \mathbb{Z}.^[32] The connected components of the complement of these affine hyperplanes are called alcoves, which are the smallest regions in this arrangement and can be viewed as bounded simplices; the fundamental alcove is the intersection of the fundamental chamber C with a suitable translate.^[32] The affine Weyl group acts simply transitively on the set of alcoves.^[32] The Weyl group W also acts on the dual space E^*, where weights \lambda reside, generating orbits W \cdot \lambda = \{ w \lambda \mid w \in W \}.^[33] These orbits partition the weight space and are finite, with the size of the orbit given by the index of the stabilizer subgroup \mathrm{Stab}_W(\lambda) = \{ w \in W \mid w \lambda = \lambda \} in W.^[33] For generic weights, the stabilizer is trivial, yielding orbits of full size |W|; for weights fixed by certain reflections, the stabilizer is larger, corresponding to parabolic subgroups of W.^[33]

Root Poset Structure

In a root system \Phi with a fixed base \Delta of simple roots, the positive roots \Phi^+ form a poset under the partial order \leq, where \alpha \leq \beta if and only if \beta - \alpha is a non-negative integer linear combination of the simple roots in \Delta.^[34] This order corresponds to the dominance partial order on the coefficient vectors of the roots when expressed in the basis \Delta.^[34] The minimal elements of this poset are precisely the simple roots themselves.^[35] The height function ht: \Phi^+ \to \mathbb{N} assigns to each positive root \alpha = \sum_{\delta \in \Delta} n_\delta \delta (with n_\delta \in \mathbb{N}) the value ht(\alpha) = \sum_{\delta \in \Delta} n_\delta, which measures the "level" of \alpha in the poset.^[35] The maximal elements, known as the maximal roots, are those \beta \in \Phi^+ such that no \gamma \in \Phi^+ satisfies \beta < \gamma, and these achieve the maximum height in the system.^[35] The partition given by the number of roots at each height level has its dual partition equal to the exponents of the associated Weyl group, as established by Kostant. The Hasse diagram of the root poset (\Phi^+, \leq) is a graded poset with rank function ht, where the rank of the minimal elements (simple roots) is 1.^[34] Covering relations in this diagram occur precisely when \alpha \prec \beta with ht(\beta) = ht(\alpha) + 1, meaning \beta = \alpha + \delta for some simple root \delta \in \Delta such that \beta is itself a root.^[34] These relations highlight the combinatorial structure, forming a directed acyclic graph where edges connect roots differing by exactly one simple root addition. This poset structure facilitates applications such as the shelling of order ideals in hyperplane arrangements associated with root systems, where the grading by height enables recursive constructions of shellable simplicial complexes.^[36] Additionally, the root poset connects to Coxeter complexes via the Weyl group orbits on roots, providing a combinatorial framework for non-crossing partitions and the topology of the associated Coxeter group actions.^[35]

Connections to Lie Theory

Root Systems in Lie Algebras

In the theory of semisimple Lie algebras over the complex numbers, root systems encode the structure of the root space decomposition \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in R} \mathfrak{g}_\alpha, where \mathfrak{h} is a Cartan subalgebra and each \mathfrak{g}_\alpha is one-dimensional. The roots R form a root system in the dual space \mathfrak{h}^*, and the interactions between root spaces are governed by the Lie bracket, which maps \mathfrak{g}_\alpha \otimes \mathfrak{g}_\beta to \mathfrak{g}_{\alpha + \beta} if \alpha + \beta \in R, or to \mathfrak{h} if \beta = -\alpha. This decomposition reveals how the algebra decomposes into irreducible representations under certain subalgebras, highlighting the pivotal role of root systems in understanding the semisimple structure.^[37] A key feature arising from this decomposition is the concept of root strings, which illustrate the finite-dimensional representations induced by individual roots. For roots \alpha, \beta \in R, the \alpha-root string through \beta consists of the consecutive roots of the form \beta + k\alpha for k \in \mathbb{Z} such that \beta + k\alpha \in R \cup \{0\}, forming an unbroken chain \beta - p\alpha, \dots, \beta + q\alpha with p, q \geq 0. The integers p and q satisfy p - q = \langle \beta, \alpha^\vee \rangle, where \alpha^\vee is the coroot, and the length of the string is p + q + 1 = 1 - \langle \beta, \alpha^\vee \rangle when \langle \beta, \alpha^\vee \rangle \leq 0. This structure ensures that the corresponding root spaces form an irreducible module under the adjoint action of a specific \mathfrak{sl}_2-subalgebra.^[38] Central to this are the \mathfrak{sl}_2-triples, which embed copies of \mathfrak{sl}_2(\mathbb{C}) into \mathfrak{g} for each root \alpha \in R. Specifically, choose basis elements e_\alpha \in \mathfrak{g}_\alpha and f_\alpha \in \mathfrak{g}_{-\alpha} such that [e_\alpha, f_\alpha] = h_\alpha, where h_\alpha = \alpha^\vee is the coroot element in \mathfrak{h} normalized so that \alpha(h_\alpha) = 2. These satisfy the relations [h_\alpha, e_\alpha] = 2e_\alpha and [h_\alpha, f_\alpha] = -2f_\alpha, making \{e_\alpha, h_\alpha, f_\alpha\} isomorphic to the standard basis of \mathfrak{sl}_2(\mathbb{C}). The subspace \mathfrak{g}_\alpha \oplus \mathfrak{g}_{-\alpha} carries the defining 2-dimensional representation of this \mathfrak{sl}_2-triple under the adjoint action, while the full algebra \mathfrak{g} decomposes into a direct sum of irreducible representations of varying dimensions determined by the root strings.^[39] Root systems also facilitate the construction of parabolic subalgebras, which are proper subalgebras containing a fixed Borel subalgebra and play a crucial role in the representation theory and geometry of semisimple Lie algebras. For a choice of simple roots \Delta \subset R, a standard parabolic subalgebra \mathfrak{p}_I corresponding to a subset I \subseteq \Delta is generated by a Borel subalgebra \mathfrak{b} = \mathfrak{h} \oplus \bigoplus_{\alpha > 0} \mathfrak{g}_\alpha together with the negative root spaces \mathfrak{g}_{-\beta} for all simple roots \beta \in I. It admits a Levi decomposition \mathfrak{p}_I = \mathfrak{l}_I \ltimes \mathfrak{u}_I, where \mathfrak{l}_I is the reductive Levi factor (the subalgebra generated by \mathfrak{h} and the root spaces for the subsystem generated by I) and \mathfrak{u}_I is the nilradical consisting of the strictly positive root spaces whose roots are not in the subsystem generated by I. This decomposition is unique up to conjugation and underscores the semidirect product structure inherent to parabolic subalgebras. Finally, root systems enable the construction of integral bases for semisimple Lie algebras, notably the Chevalley basis, which provides a canonical integer structure. This basis consists of elements \{x_\alpha \mid \alpha \in R\} \cup \{h_i \mid i = 1, \dots, r\}, where r = \dim \mathfrak{h} is the rank, x_\alpha are chosen in \mathfrak{g}_\alpha such that [x_\alpha, x_{-\alpha}] = h_\alpha for positive roots \alpha, and the h_i form a basis for the coroot lattice. All Lie brackets in this basis have structure constants in \mathbb{Z}, ensuring the basis spans a \mathbb{Z}-lattice stable under the bracket and facilitating constructions over rings of integers, such as in the theory of Chevalley groups.^[40]

Cartan Subalgebras and Roots

In the theory of semisimple Lie algebras over the complex numbers, a Cartan subalgebra \mathfrak{h} of a Lie algebra \mathfrak{g} is defined as a maximal toral subalgebra, meaning it is an abelian subalgebra on which the adjoint representation is simultaneously diagonalizable, or equivalently, \mathfrak{h} is nilpotent and equals its normalizer N_{\mathfrak{g}}(\mathfrak{h}) = \{ x \in \mathfrak{g} \mid \mathrm{Ad}(x) \mathfrak{h} = \mathfrak{h} \}.^[41] This structure ensures that \mathfrak{h} consists of ad-semisimple elements, and every semisimple Lie algebra possesses such a subalgebra, with the dimension of \mathfrak{h} being the rank of \mathfrak{g}.^[42] A fundamental property is that all Cartan subalgebras of \mathfrak{g} are conjugate under the adjoint action of the corresponding Lie group, implying they share the same dimension and structural features.^[43] Roots arise naturally from the adjoint action of \mathfrak{h} on \mathfrak{g}. Specifically, the Lie algebra decomposes as \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta} \mathfrak{g}_{\alpha}, where \Delta \subset \mathfrak{h}^* is the root system consisting of nonzero linear functionals \alpha: \mathfrak{h} \to \mathbb{C} such that the root spaces \mathfrak{g}_{\alpha} = \{ x \in \mathfrak{g} \mid [\mathfrak{h}, x] = \alpha(\mathfrak{h}) x \ \forall h \in \mathfrak{h} \} are nonzero.^[43] On each root space, the adjoint action satisfies \mathrm{ad}_h \big|_{\mathfrak{g}_{\alpha}} = \alpha(h) \cdot \mathrm{id} for all h \in \mathfrak{h}, reflecting the eigenvalue character of the decomposition.^[44] The dual space \mathfrak{h}^* is canonically identified with \mathfrak{h} via the Killing form B(X,Y) = \mathrm{tr}(\mathrm{ad}_X \mathrm{ad}_Y), which restricts to a nondegenerate symmetric bilinear form on \mathfrak{h}, allowing roots to be viewed as elements of \mathfrak{h} itself.^[45] The nilpotent radical emerges in the context of parabolic subalgebras, particularly Borels. A Borel subalgebra \mathfrak{b} is a maximal solvable subalgebra containing \mathfrak{h}, decomposed as \mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}, where \mathfrak{n} = \bigoplus_{\alpha > 0} \mathfrak{g}_{\alpha} is the sum over positive roots with respect to a choice of Weyl chamber, and \mathfrak{n} forms the nilpotent radical of \mathfrak{b} since [\mathfrak{n}, \mathfrak{n}] \subset \mathfrak{n} with higher nilpotency class.^[46] Regular elements in \mathfrak{h} are those h \in \mathfrak{h} such that \alpha(h) \neq 0 for all roots \alpha \neq 0, ensuring the centralizer C_{\mathfrak{g}}(h) = \{ x \in \mathfrak{g} \mid [h, x] = 0 \} = \mathfrak{h} has minimal dimension equal to the rank of \mathfrak{g}.^[47] These elements are dense in \mathfrak{h}, and their adjoint orbits generate the Cartan subalgebra under the group action, highlighting the stability of the toral structure.^[48]

Explicit Realizations of Irreducible Systems

Series A_n

The root system of type A_n (for n \geq 1) is one of the classical irreducible root systems, realized explicitly in the n-dimensional hyperplane

H = \left\{ x \in \mathbb{R}^{n+1} \;\middle|\; \sum_{i=1}^{n+1} x_i = 0 \right\}

of \mathbb{R}^{n+1}, equipped with the restriction of the standard Euclidean inner product. The roots are the vectors \alpha_{ij} = e_i - e_j for all distinct indices $1 \leq i, j \leq n+1, where \{e_1, \dots, e_{n+1}\} denotes the standard orthonormal basis of \mathbb{R}^{n+1}. These roots span H and satisfy the defining axioms of a root system, including closure under reflections across root hyperplanes.^[1]^[20] A standard choice of positive roots consists of those \alpha_{ij} with i < j, of which there are \binom{n+1}{2} = \frac{n(n+1)}{2}; the full set of roots then comprises these positive roots together with their negatives, yielding a total of n(n+1) roots. The simple roots forming a basis for this positive system are \alpha_i = e_i - e_{i+1} for i = 1, \dots, n. With respect to the induced inner product on H, every root has the same squared length \langle \alpha_{ij}, \alpha_{ij} \rangle = 2, so the length is \sqrt{2}; this uniformity classifies A_n as a simply-laced root system.^[1]^[20] The Weyl group W(A_n) is generated by reflections across the hyperplanes perpendicular to the simple roots and is isomorphic to the symmetric group S_{n+1}, acting on H by permuting the coordinates of vectors in \mathbb{R}^{n+1}. The Dynkin diagram of A_n is a linear chain of n nodes connected by single (unoriented) bonds, reflecting the equal angles between adjacent simple roots. This root system arises naturally as the root system of the semisimple complex Lie algebra \mathfrak{sl}(n+1, \mathbb{C}), consisting of trace-zero (n+1) \times (n+1) matrices.^[1]^[20]^[49]

Series B_n and C_n

The root system B_n (for n \geq 2) is realized in the Euclidean space \mathbb{R}^n equipped with the standard dot product. Its roots consist of the short roots \pm e_i (for $1 \leq i \leq n) and the long roots \pm e_i \pm e_j (for $1 \leq i < j \leq n), where \{e_1, \dots, e_n\} is the standard orthonormal basis.^[50] The simple roots are \alpha_i = e_i - e_{i+1} for $1 \leq i \leq n-1 and \alpha_n = e_n.^[50] Under this realization, the short roots have length \sqrt{1} = 1, while the long roots have length \sqrt{2}.^[50] The root system C_n (for n \geq 3) shares the same ambient space \mathbb{R}^n but features long roots \pm 2e_i (for $1 \leq i \leq n) and short roots \pm e_i \pm e_j (for $1 \leq i < j \leq n).^[50] Its simple roots are \alpha_i = e_i - e_{i+1} for $1 \leq i \leq n-1 and \alpha_n = 2e_n.^[50] Here, the short roots have length \sqrt{2}, and the long roots have length $2, reversing the length disparity of B_n.^[50] The systems B_n and C_n are dual to each other, satisfying \Phi(C_n) = \Phi^\vee(B_n), where \Phi^\vee denotes the dual root system obtained by rescaling roots inversely to their lengths.^[4] Both B_n and C_n share the same Weyl group, which is the hyperoctahedral group of signed permutations on n elements, isomorphic to (\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n and of order $2^n n!.^[10] This group is generated by reflections across the hyperplanes perpendicular to the roots. The Dynkin diagram for B_n consists of a chain of n-1 single edges connecting n nodes, followed by a double edge with an arrow pointing toward the final node, indicating the shorter root there.^[50] For C_n, the diagram is similar but with the arrow on the double edge pointing away from the final node, reflecting the longer root at that end.^[50] These diagrams encode the off-diagonal entries of the Cartan matrix via the angle between adjacent simple roots, with the arrow denoting the distinction in root lengths.^[50]

Series D_n

The root system D_n (for n \geq 4) is an irreducible simply laced root system of rank n, corresponding to the Lie algebra \mathfrak{so}(2n, \mathbb{C}) of the special orthogonal group in even dimension. It consists of $2n(n-1) roots, all of equal length \sqrt{2}, embedded in the Euclidean space \mathbb{R}^n equipped with the standard dot product and orthonormal basis \{e_1, \dots, e_n\}. The roots are explicitly given by all vectors of the form \pm e_i \pm e_j for $1 \leq i < j \leq n. A standard choice of positive roots comprises the n(n-1) vectors e_i + e_j (with i < j) and e_i - e_j (with i < j); among these, the two maximal roots are e_{n-1} + e_n and e_{n-1} - e_n. The subsystem consisting of the differences e_i - e_j ( i \neq j ) forms an A_{n-1} root system. The simple roots are \alpha_k = e_k - e_{k+1} for $1 \leq k \leq n-2, \alpha_{n-1} = e_{n-1} - e_n, and \alpha_n = e_{n-1} + e_n; these form a basis for \mathbb{R}^n and span the positive roots as nonnegative integer combinations. The associated Dynkin diagram is a linear chain of n-2 nodes connected by single edges, with the (n-2)-th node branching into two additional nodes (labeled n-1 and n) via single edges, reflecting the equal lengths and angles of $120^\circ between the branching roots. The Weyl group W(D_n) is the semidirect product (\mathbb{Z}/2\mathbb{Z})^{n-1} \rtimes S_n, consisting of all permutations of the coordinates together with an even number of sign flips; it has order $2^{n-1} n! and acts faithfully on \mathbb{R}^n by preserving the root system. This group is realized as the quotient of the normalizer of a maximal torus in \mathrm{SO}(2n) by the torus itself.

Exceptional Systems E_6, E_7, E_8, F_4, G_2

The exceptional root systems consist of five irreducible systems—G₂, F₄, E₆, E₇, and E₈—that do not belong to the classical series Aₙ, Bₙ, Cₙ, or Dₙ and exhibit unique symmetries and structures.^[51] These systems are finite sets of vectors in Euclidean spaces satisfying the axioms of root systems, with the E-series being simply laced (all roots of equal length) while F₄ and G₂ feature roots of two different lengths.^[51] Their Dynkin diagrams, which encode the simple roots and their inner products via standard conventions (single bonds for angle 120°, double for 135°, triple for 150°, and arrows indicating length differences), are indecomposable and non-classical.^[51] The Weyl groups, generated by reflections across the hyperplanes perpendicular to the roots, have orders that reflect the high degree of symmetry in these systems.^[51] The smallest exceptional system, G₂, is of rank 2 and realized in the plane orthogonal to (1,1,1) within ℝ³.^[51] It comprises 12 roots: six short roots forming a regular hexagon with vertices at 60° intervals, such as ±(e₁ - e₂), ±(e₂ - e₃), and ±(e₃ - e₁), and six long roots of length √6 (√3 times the short root length √2), pointing in directions like ±(2e₁ - e₂ - e₃), ±(2e₂ - e₃ - e₁), and ±(2e₃ - e₁ - e₂).^[20] A standard choice of simple roots is the short root α₁ = e₁ - e₂ and the long root α₂ = 2e₂ - e₁ - e₃ (up to Weyl action). The Dynkin diagram for G₂ is two nodes connected by a triple bond with an arrow from the long to the short root, indicating the length ratio of √3:1 between long and short roots.^[20] The Weyl group of G₂ is the dihedral group of order 12, acting as rotations and reflections preserving the hexagonal symmetry.^[51] Next, F₄ is the rank-4 exceptional system in ℝ⁴ with 48 roots, mixing short and long lengths in a 1:√2 ratio.^[51] The roots consist of 24 long roots of the form ±eᵢ ± eⱼ (1 ≤ i < j ≤ 4) and 24 short roots: the 8 vectors ±e_i (i=1 to 4) and all 16 vectors ½(±1, ±1, ±1, ±1) with arbitrary sign combinations, all of length 1.^[52] A standard choice of simple roots includes two short and two long, such as α₁ = (1/2, -1/2, -1/2, -1/2), α₂ = e₄, α₃ = e₃ - e₄, α₄ = e₂ - e₃. Its Dynkin diagram features four nodes in a chain: a single bond, followed by a double bond with an arrow pointing right (indicating the short root on the right), and another single bond.^[51] The Weyl group has order 1152 and includes the hyperoctahedral group as a subgroup, reflecting the coordinate permutations and sign changes.^[51] The E-series systems are simply laced and embedded in higher dimensions with branching Dynkin diagrams. E₆, of rank 6 in ℝ⁶, has 72 roots and a diagram consisting of a chain of five nodes with a sixth node branching from the third.^[51] Its Weyl group order is 51,840.^[51] Simple roots can be given explicitly in coordinates, but are more involved; one realization uses an orthogonal complement in ℝ⁷ or standard basis vectors adjusted for the diagram. E₇, rank 7 in ℝ⁷, contains 126 roots, with a diagram extending the E₆ chain by one more node and the branch from the third position; the Weyl group order is 2,903,040.^[51] E₈, the largest at rank 8 in ℝ⁸, has 240 roots and a self-dual root system (invariant under duality), with a diagram that further extends the chain to seven nodes plus the branch from the third.^[51] The Weyl group order for E₈ is 696,729,600.^[51] One explicit realization of the E₈ roots involves coordinates in ℝ⁸ using the even-coordinate D₈ lattice plus half-integer spinor vectors, and alternative constructions project onto lower dimensions incorporating the golden ratio φ = (1 + √5)/2 in certain basis elements to capture the exceptional symmetry.^[51]^[53]

References

[1]
[PDF] Classification of root systems
Sep 8, 2017 · Definition 1.5. Suppose V is a finite-dimensional real vector space and V ∗ is its dual space. A vector root system consists of a finite set R ...
[2]
[PDF] The classification of root systems
Nov 28, 2007 · November 28, 2007. 1 Introduction. An irreducible root system is a finite set of vectors in Euclidean space satisfying certain properties.
[3]
[PDF] Introduction to Lie Algebras and Representation Theory (following ...
Jun 6, 2020 · Remember that an isomorphism of root systems preserves the Cartan integers (by definition). So, the Cartan integers are some discrete ...
[4]
[PDF] 18.745 F20 Lecture 21: Root Systems - MIT OpenCourseWare
A root system R is reduced if for α, cα ∈ R, we have c = ±1. Proposition 21.3. If g is a semisimple Lie algebra and h ⊂ g a Car- tan subalgebra then the ...
[5]
[PDF] classification of root systems - UChicago Math
2. Root systems. Humphreys defines a root system as a subset Φ of the Euclidean space E which. satisfies the following properties: (1) Φ is finite and spans E, ...Missing: mathematics | Show results with:mathematics
[6]
[PDF] Lecture 21 — The Weyl Group of a Root System
Nov 23, 2010 · The group W is called the Weyl group of the root system (V,∆) (and of the corresponding semisimple lie algebra g). Proposition 21.1.
[7]
[PDF] Lie algebras - Harvard Mathematics Department
Apr 23, 2004 · system, then all roots of the same length are conjugate under the Weyl group. ... The Weyl group Wg acts simply transitively on the Weyl chambers ...
[8]
[PDF] CRYSTALLOGRAPHIC ROOT SYSTEMS
This project shows how the attempt to classify all semisimple Lie algebras in an alge- braically closed field of characteristic 0 led to the development of ...
[9]
[PDF] What is the order of the Weyl group of E8? We'll do this by 4 different
So the order of the Weyl group is. 135 · 27 · 8! = |27 · S8| ×. # norm 4 elements. 2 × dim L. Remark 312 Conway used a similar method to calculate the order of ...
[10]
[PDF] root systems - george h. seelinger
Given Cartan matrix M of a root system Φ,. (a) The transpose of the Cartan matrix expresses the simple roots as linear combinations of the fundamental weights.
[11]
None
### Summary of Positive Roots, Simple Roots, and Cartan Matrix in Root Systems
[12]
[PDF] Basics of Lie theory - Classification of Lie Algebras - ETH Zürich
Mar 11, 2012 · the rank of the Lie algebra. rank 1 There is exactly one possible root system that can be drawn. (A1). This is precisely the root system of sl2C ...
[13]
[PDF] Root systems
Another root system on V = R is {±α,±2α}. It is called BC1. We now assume an inner product defined on V , standard inner product, because it would then be easy ...
[14]
[PDF] LECTURE 10 8 June 2010 1. Irreducible root systems Let E be a ...
Jun 8, 2010 · Lemma 1.2. A root system ∆ is irreducible if and only if Π can not be partitioned into the union of two proper subsets Π = Π1 ∪ Π2, ...
[15]
[PDF] paws 2024: symmetries of root systems - Arizona Winter School
Every rank two root system is isomorphic to A1 ˆ A1,A2,B2, or G2. Proof. We ... Every root system is the direct sum of some set of irreducible root systems,.
[16]
[PDF] The classification theorem
Apr 27, 2020 · Examples in rank 2. Here are the three irreducible root systems of rank 2: A2. Dynkin diagram: B2. Dynkin diagram: (n ≥ 1). (n ≥ 2). E. G2.
[17]
[PDF] On Generalized Root Systems - arXiv
Oct 16, 2024 · ... define the Weyl group of R as the subgroup of the symmetric group of R generated by all σα. If R is a root system, then this group is isomorphic.
[18]
Representations of Rank 2 Reductive Groups - Joel Gibson
Feb 9, 2020 · The other root systems A_2, B_2, G_2 and BC_2 are called irreducible since they cannot be decomposed in this way. (Be careful not to confuse “ ...
[19]
[PDF] Complex Semisimple Lie Algebras - Pierre Clare
Every complex root system can be obtained in the above way. More precisely: Theorem 5'. Let R be a root system in a complex vector space V. Let V0 be the. R ...
[20]
[PDF] root systems and dynkin diagrams - Cornell Mathematics
A root system Φ is a set of vectors in Rn such that: (1) Φ spans Rn and 0 6∈ Φ. (2) If α ∈ Φ and λα ∈ Φ, then λ = ±1.
[21]
[PDF] Introduction to Lie Algebras and Representation Theory
(5) Root systems are treated axiomatically (Chapter III), along with some of ... This means that any simple Lie algebra is isomorphic to a linear Lie algebra.
[22]
[PDF] Semisimple Lie Algebras and the Root Space Decomposition
May 1, 2015 · This document will develop just the material needed to describe a semisimple Lie algebra in terms of its root space decomposition.
[23]
[PDF] Lecture 2: Finite-dimensional simple Lie algebras
The affine Weyl group. Central Extensions. The dual root system. In addition to the root system Φ ⊂ h∗ we have a dual root system Φ. ∨. ⊂ h. Indeed there is ...
[24]
[PDF] 18 Root systems - mimuw
In this chapter, we are concerned with the theory of root systems. This simple geometric notion will play an amazingly important role in the structure ...
[25]
[PDF] Lecture 16 — Root Systems and Root Lattices
Nov 1, 2010 · Corollary 16.3. If (V,∆) is a root system, then Q := Z∆ is a lattice, called the root lattice. Proof. By the two propositions, the corollary ...
[26]
[PDF] Math 210C. Size of fundamental group 1. Introduction Let (V,Φ) be a ...
It certainly has to contain the root lattice Q := ZΦ determined by the root system (V,Φ). Exercise 1(iii) in HW7 computes the index of this inclusion of ...
[27]
[PDF] Lecture 17 — Cartan Matrices and Dynkin Diagrams
Nov 4, 2010 · Note that when the two roots in question have different lengths, the arrow points to the shorter root. The diagram formed in this way is the ...
[28]
[PDF] simple roots, cartan matrices and dynkin diagrams
We define Cartan matrix and Dynkin diagram associated to a root system and state the Classification theorem to be proven next week. The material covered can be ...
[29]
[PDF] 18.745 F20 Lecture 23: Dynkin Diagrams - MIT OpenCourseWare
Classification of Dynkin diagrams. The following theorem gives a complete classification of irreducible root systems. Theorem 23.7. (i) Connected Dynkin ...Missing: Killing | Show results with:Killing
[30]
[PDF] Reference sheet for classical roots systems - UC Berkeley math
Sep 21, 2023 · Mnemonic: “sln is the first semisimple Lie algebra you ever learn about, so its Dynkin diagram comes first in the alphabet.” • Weyl group: Sn.
[31]
[PDF] 18 Root systems and reflection groups - UC Berkeley math
An example of a non-reduced root system is BCn, consisting of the vectors ±xi, ±xi ± xj, and. ±2xi. In fact these are the only irreducible non-reduced root ...
[32]
[PDF] Affine Weyl group alcoves - MIT Mathematics
Oct 12, 2022 · f. W = W n T = affine Weyl group. Everything below works for W = any Weyl group, T = root lattice. An alcove is a conn component of ...
[33]
[PDF] SU(n), Weyl Chambers and the Diagram of a Group 1
The Weyl group acts on the set of roots, but does not respect the decomposition into positive and negative roots. A specific example to continually keep in mind ...
[34]
[PDF] arXiv:1410.0195v1 [math.CO] 1 Oct 2014
Oct 1, 2014 · The root poset on Φ+ is the partial order defined by γ ≤ γ/ if and only. if γα ≤ γ/ α for all α ∈ ∆.
[35]
[PDF] The root posets. Four lectures (Sanya Workshop 2014)
Root posets play a decisive role in many parts of mathematics: of course in Lie theory, in geometry (hyperplane arrangements) and group theory (reflection ...
[36]
[PDF] Shellability of posets of labeled partitions and arrangements defined ...
Jul 13, 2017 · We prove that the posets of connected components of intersections of toric and elliptic arrangements defined by root systems are EL-shellable ...
[37]
[PDF] Root systems and semisimple complex Lie algebras
Apr 29, 2020 · Root systems are classified by Dynkin diagrams, which determine the root system. These systems are related to classical Lie algebras, and the ...
[38]
[PDF] root strings - Brandeis
Claim: Lα+β ⊕ Vβ ∼= V (1). Pf: Since xα ∈ Lα, adxα sends Lα+β to L2α+β = 0. Therefore, x13 = xα+β is a maximal vector. But hα(x13) = (h1 − h3)x13 = x13.
[39]
[PDF] LECTURE 11. Proposition 1.1. Let g be a semisimple Lie algebra ...
A triple of elements {e,f,h} in a Lie algebra g which obey the relations of the standard generators of sl2 (that is, [e, f] = h, [h, e]=2e, [h, f]=2f) is called.
[40]
[PDF] tits systems, parabolic subgroups, parabolic subalgebras
Levi Decompositions PI as a semidirect product of LV is called a Levi decomposition, and we call L a Levi factor. In general, such a decomposition exists in an ...
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[PDF] semisimple lie algebras and the chevalley group construction
Abstract. This paper illustrates the construction of a Chevalley group for a finite dimensional semisimple Lie algebra over an algebraically complete field.
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[PDF] Lecture 8 — Cartan Subalgebra
Oct 5, 2010 · Definition 8.2. A Cartan subalgebra of a Lie algebra g is a subalgebra h, satisfying the following two conditions: i) h is a nilpotent Lie ...
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[PDF] Introductory Lie Theory Notes - UC Berkeley math
1.1 Definitions, Lie subgroups, and the closed subgroup theorem. Definition 1.1. A real Lie group is a group G such that G is also a manifold and the group.
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[PDF] Cartan subalgebras and root decomposition
Apr 13, 2020 · Since sl(n,C) is simple, any two invariant bilinear forms are proportional. The Killing form must therefore be a multiple of the trace pairing.
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[PDF] Lie Groups: Fall, 2022 Lecture VI Structure of Simple Lie Algebras
Nov 28, 2022 · 1.1 The A-series An = sl(n + 1,C) for n ≥ 1. One denotes by An,n ≥ 1, the Lie algebra sl(n + 1,C). This is a Lie alebra.Missing: A_n | Show results with:A_n
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[PDF] Lecture 13 — Structure Theory of Semisimple Lie Algebras II
Oct 21, 2010 · The Killing form K of g is non-degenerate. 2. The algebra g contains a Cartan subalgebra h. Furthermore, h is abelian and diagonalizable on g, ...
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[PDF] arXiv:0908.4201v1 [math.RT] 28 Aug 2009
Aug 28, 2009 · In this paper, we give upper bounds for the index of the quotient of the Borel subalgebra of a simple Lie algebra or its nilpotent radical by an ...
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[PDF] Theorem 2.9. Any finite-dimensional complex Lie algebra g has a ...
Cartan subalgebra. Before coming to the proof, we introduce “regular” elements of g. In sl(n, C) the regular elements will be the matrices with distinct ...<|control11|><|separator|>
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[PDF] Lecture 12 — Structure Theory of Semisimple Lie Algebras (I)
Oct 19, 2010 · 0 ⊃ Rw is a Cartan subalgebra. So gw. 0 = Rw, implying Rw is Cartan. Conversely, if Rw is Cartan, then w is regular. Otherwise w would be.
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http://virtualmath1.stanford.edu/~conrad/249BW16Page/handouts/classicalgps.pdf
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[PDF] Math 249B. Root systems for split classical groups
In this handout, we work out the root systems for the split connected semisimple groups SLn+1 (n ≥ 1), SO2n+1 (n ≥ 2), Sp2n (n ≥ 2), and SO2n (n ≥ 3) over an ...
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[PDF] Root systems and Weyl groups
Sep 11, 2002 · Thus W acts on R. In the case of a root system coming from a group the Weyl group is isomorphic to NormG(T)/T. Example ...
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[PDF] A new construction of E8 and the other exceptional root systems
2(1 +p5) ⇡ 1.618 (golden ratio). T. G. Looks like a virus or carbon onion. Pierre-Philippe Dechant. A new construction of E8 and the other exceptional root ...Missing: coordinates | Show results with:coordinates