Cartan subalgebra

For a complex semisimple Lie algebra, a Cartan subalgebra is a maximal abelian subalgebra consisting of semisimple elements. It is also called a maximal torus in this setting.

For a Cartan subalgebra $H$ of a finite-dimensional complex semisimple Lie algebra $L$, $H$ is abelian and $C_L(H)=H$, so it is maximal among abelian subalgebras. Starting from the nilpotent self-normalizing definition, the generalized-weight decomposition for a nilpotent Lie algebra and the Engel theorem show that the zero generalized weight space of its adjoint action is exactly $H$. Invariance of the Killing form makes $H$ orthogonal to every nonzero generalized weight space. The Lie theorem gives $B([H,H],H)=0$, and nondegeneracy then forces $[H,H]=0$. An element commuting with $H$ normalizes it, hence belongs to $H$.

For a complex semisimple Lie algebra, the Cartan subalgebra is orthogonal under the Killing form to every nonzero root space. A vector in the Cartan subalgebra orthogonal to that whole subalgebra is therefore orthogonal to the entire Lie algebra and must vanish. On the Cartan subalgebra the form is $\kappa (h,h^{\prime })={\sum }_{\alpha \in \Phi }\alpha (h)\alpha (h^{\prime })$.

The real span of the simple coroots inside a complex Cartan subalgebra carries a positive-definite restriction of the Killing form. Equivalently it consists of the $h$ for which all roots have real values. The formula $\kappa (h,h)={\sum }_{\alpha \in \Phi }\alpha (h{)}^2$ is positive for $h\ne 0$, since the roots span the dual. The induced dual inner product makes the root system Euclidean and makes root reflections orthogonal.

Let $\mathcal{B}(H)=B(H,\cdot )$ be the Killing form identification of a Cartan subalgebra with its dual. The root-space decomposition gives $B(H,K)={\sum }_{\beta \in \Phi }\beta (H)\beta (K)$, and therefore the displayed root expansion. On the real coroot span its coefficients are real, proving $\mathcal{B}({\mathfrak{h}}_{\mathbb{R}})\subseteq \mathbb{R}P$. For ${\mathfrak{sl}}_3$ and $H_{12}=E_{11}-E_{22}$, the formula gives $\mathcal{B}(H_{12})=6(L_1-L_2)$. The normalization is $B(X,Y)=\mathrm{tr}(\mathrm{ad}X\mathrm{ad}Y)$, not a rescaled trace form.

// Target: semisimple-lie-algebra.bigb

The rank is the dimension of a Cartan subalgebra, equivalently the dimension of the real span of its root system.

For a Cartan subalgebra $\mathfrak{t}$, a semisimple Lie algebra decomposes asThe nonzero functionals $\alpha$ are the roots.

Choosing a positive system of a root system for a complex semisimple Lie algebra divides its root spaces into negative and positive ones. Their respective sums ${\mathfrak{n}}^{-}$ and ${\mathfrak{n}}^{+}$, together with the Cartan subalgebra $\mathfrak{h}$, give this direct sum of vector spaces. Both ${\mathfrak{n}}^{\pm }$ are nilpotent Lie algebras, and $\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ is a Borel subalgebra. The Poincaré-Birkhoff-Witt theorem turns this into an ordered vector-space decomposition of the universal enveloping algebra, useful for building Verma modules.

For a complex semisimple Lie algebra, restrict the module $\mathfrak{h}\oplus {⨁}_{j\ne 0}{\mathfrak{g}}_{j\alpha }$ to the sl2 subalgebra associated with a root. All its weights are even. Its raising operator maps the weight-zero space $\mathfrak{h}$ onto the one-dimensional line of the chosen root vector, so there is exactly one nontrivial irreducible summand: the adjoint module of highest weight $2$. Thus the root spaces at $\pm \alpha$ are one-dimensional and $2\alpha$ is not a root. For any parallel root $c\alpha$, the integers $2c$ and $2/c$ have product $4$; excluding double and half roots leaves only $c=\pm 1$.

For a root $\alpha$, its root space is the simultaneous eigenspaceof the Adjoint representation of the Cartan subalgebra.

For root spaces ${\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }$, the Jacobi identity gives $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]\subset {\mathfrak{g}}_{\alpha +\beta }$ when $\alpha +\beta \ne 0$. If the bracket is nonzero, that nonzero weight must be a root. When $\beta =-\alpha$, the bracket instead lies in the Cartan subalgebra and can be nonzero; zero is not a root. Thus the common nonzero-bracket implication requires the opposite-root exception.

A root vector is a nonzero element of a root space ${\mathfrak{g}}_{\alpha }$ in the root-space decomposition of a semisimple Lie algebra.

An element $x$ of a semisimple Lie algebra of rank $\ell$ is regular when its centralizer has the minimum possible dimension $\ell$.

For $t$ in a Cartan subalgebra $\mathfrak{t}$, the root-space decomposition givesThus $t$ is regular exactly when $\alpha (t)\ne 0$ for every root $\alpha$.

Let a maximal torus $\mathfrak{t}$ give a decomposition $\mathfrak{g}={\mathfrak{g}}_{\alpha }\oplus \mathfrak{t}\oplus {\mathfrak{g}}_{\beta }$. If $\mathfrak{g}$ has a nondegenerate trace form, then $\beta =-\alpha$, the opposite root spaces generate ${\mathfrak{sl}}_2$, andwhere $\ker \alpha \subset \mathfrak{t}$ is central and abelian.

For every root $\alpha$ of a complex semisimple Lie algebra, there are root vectors $e_{\alpha }\in {\mathfrak{g}}_{\alpha }$ and $f_{\alpha }\in {\mathfrak{g}}_{-\alpha }$ and an element $h_{\alpha }\in \mathfrak{t}$ satisfyingTheir span is an ${\mathfrak{sl}}_2$ subalgebra.

For a root of a root system of a complex semisimple Lie algebra, define $t_{\alpha }$ by $B(t_{\alpha },h)=\alpha (h)$ using the Killing form on its Cartan subalgebra. Then $\alpha (t_{\alpha })\ne 0$. Otherwise vectors $x\in {\mathfrak{g}}_{\alpha }$, $y\in {\mathfrak{g}}_{-\alpha }$ with $B(x,y)=1$ would generate a solvable Lie algebra with $[x,y]=t_{\alpha }$ central. The Lie theorem would make $\mathrm{ad}{t}_{\alpha }$ nilpotent, while the root-space decomposition makes it diagonalizable, forcing $t_{\alpha }$ into the zero center of a Lie algebra.

A root system is a finite set of nonzero vectors closed under the reflections they define and satisfying the crystallographic integrality condition when it arises from a semisimple Lie algebra.

The hyperplane perpendicular to a root of a root system is its root hyperplane, interpreted as the kernel of the associated linear functional on the Cartan space. The corresponding reflection is $s_{\alpha }(v)=v-2⟨v,\alpha ⟩\alpha /⟨\alpha ,\alpha ⟩$. For the B2 root system, the four distinct hyperplanes are the two coordinate axes and the two diagonals; their reflections generate the signed-permutation Weyl group of order eight.

A root subsystem of a root system $\Phi$ is a subset that is itself a root system in its real span. In particular, if a subspace $F$ is spanned by roots, $\Phi \cap F$ is a full root subsystem: root reflections in its roots preserve both $\Phi$ and $F$. Reflection-generated subsystems need not contain every ambient root in their span. A pair is a simple root basis only when every subsystem root has integral coordinates of one sign in that pair; merely choosing two independent roots is insufficient.

After choosing positive roots in a root system, a negative root is the negative of a positive root. Equivalently, for a regular linear functional defining the positive roots, its value on a negative root is negative. Relative to the associated simple roots, the coefficients of a negative root are all nonpositive integers.

For nonproportional roots in a reduced root system which is a crystallographic root system, the negative Cartan integers $m=2(\alpha ,\beta )/(\alpha ,\alpha )$ and $n=2(\alpha ,\beta )/(\beta ,\beta )$ satisfy $mn=4{\cos }^2\theta \in \{1,2,3\}$. At least one of $m,n$ equals $-1$, so reflecting one root in the other produces their sum. The exclusion of opposite roots is essential: their inner product is negative but their sum is zero, which is not a root.

A root system spanning a real inner product space is irreducible when it cannot be partitioned into two nonempty mutually orthogonal root systems. Its Dynkin diagram is connected. This condition concerns the roots themselves, rather than an arbitrary ambient space larger than their span.

For an irreducible root system spanning $V$, the span $U$ of a nonzero root's Weyl group orbit is invariant, as is $U^{\perp }$. A root reflection preserving both subspaces must have its normal entirely in one of them: otherwise its rank-one correction mixes the subspaces. Thus a proper nonzero $U$ would partition all roots into mutually orthogonal subsets, contradicting irreducibility. Consequently some orbit root is nonorthogonal to any given root. Since Weyl reflections preserve lengths, the pairwise integrality restrictions for nonorthogonal roots also bound length ratios of orthogonal roots. All squared-length ratios lie in $\{1,2,1/2,3,1/3\}$; there cannot be three distinct lengths, since ratios two and three relative to a shortest root would yield the forbidden ratio $3/2$.

A finite root system in a real inner product space is crystallographic when every Cartan integer $2(\beta ,\alpha )/(\alpha ,\alpha )$ is an integer. The roots of a complex semisimple Lie algebra satisfy this because each number is a weight of its sl2 subalgebra associated with a root. Reducedness is a separate axiom; the root-space reducedness lemma establishes it for these Lie-algebra roots.

A root is an individual nonzero vector in a root system. For a complex semisimple Lie algebra, it is a nonzero weight of the Adjoint representation restricted to a Cartan subalgebra; its root space consists of vectors $x$ satisfying $[h,x]=\alpha (h)x$ for all $h$ in that subalgebra.

A reduced root system of rank one is $\Phi =\{\alpha ,-\alpha \}$. Its fundamental weight is $\alpha /2$, and its Weyl group has two elements. Its ordinary Dynkin diagram has one node; adjoining ${\alpha }_0=-\alpha$ gives affine Cartan matrix $\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$, drawn as a double bond between equal-length roots.

The $A_2$ root system has equal-length simple roots ${\alpha }_1,{\alpha }_2$ with angle $2\pi /3$, and roots $\pm {\alpha }_1$, $\pm {\alpha }_2$, $\pm ({\alpha }_1+{\alpha }_2)$. Its Cartan matrix is $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$. The corresponding complex simple Lie algebra is ${\mathfrak{sl}}_3(\mathbb{C})$, of dimension eight.

For the A2 root system, ${\omega }_1=(2{\alpha }_1+{\alpha }_2)/3$ and ${\omega }_2=({\alpha }_1+2{\alpha }_2)/3$. The weight lattice $P=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ contains the root lattice $Q$ with index three. In fundamental coordinates $(a,b)$, root-lattice membership is $a-b\equiv 0(\mathrm{mod}3)$.

An automorphism of a root system $\Phi \subset E$ is an orthogonal transformation of $E$ mapping $\Phi$ onto itself. The Weyl group is a subgroup of this automorphism group; additional automorphisms can act nontrivially on a Dynkin diagram while preserving a root basis.

In an irreducible root system with two root lengths, a long root has the larger length with respect to a Weyl group invariant inner product.

In an irreducible root system with two root lengths, a short root has the smaller length with respect to a Weyl group invariant inner product.

For a nonzero vector $\alpha$ in a Euclidean normed vector space, the root reflection isIt fixes the hyperplane perpendicular to $\alpha$ and sends $\alpha$ to $-\alpha$.

The Cartan integer associated with two roots isA crystallographic root system requires every Cartan integer to be an integer.

A root system is reduced when the only scalar multiples of any root $\alpha$ that are also roots are $\alpha$ and $-\alpha$.

The $A_3$ root system is $\{e_i-e_j:1\le i\ne j\le 4\}$ in the hyperplane of ${\mathbb{R}}^4$ whose coordinates sum to zero. It has $12$ roots and simple roots $e_1-e_2,e_2-e_3,e_3-e_4$.

The Cartan matrix of type A3 is

A root string is the set $S_{\alpha ,\beta }=\Phi \cap (\beta +\mathbb{Z}\alpha )$ for roots $\alpha ,\beta$ of a root system $\Phi$. For nonproportional roots $\alpha$ and $\beta$ of a reduced crystallographic root system, the roots of the form $\beta +n\alpha$ form an unbroken root string $\beta -p\alpha ,\dots ,\beta +q\alpha$, and $p-q=⟨\beta ,{\alpha }^{\vee }⟩$. The root-string theorem explains the endpoint relation. If $\beta =\pm \alpha$, zero is missing and this consecutive-string statement does not apply.

For nonproportional roots $\gamma ,\delta$ of a reduced crystallographic root system, $S_{\gamma ,\delta }=\{\delta -p\gamma ,\dots ,\delta +q\gamma \}$ is consecutive, and $p-q=⟨\delta ,{\gamma }^{\vee }⟩$. Realize the crystallographic root system as the roots of a complex semisimple Lie algebra. The sum of the one-dimensional root spaces in this string is a module for the sl2 subalgebra associated with a root. Its weights are $⟨\delta ,{\gamma }^{\vee }⟩+2n$; they have one fixed parity. Complete reducibility gives a sum of irreducible representations of the sl2 Lie algebra. Two summands of this parity would overlap at weight zero when the parity is even, or at weights $\pm 1$ when it is odd, contradicting the one-dimensional root spaces. Thus there is one irreducible string, whose opposite endpoint weights sum to zero. This proves the relation. For distinct simple roots, $p=0$ and the cardinality is $1-⟨\delta ,{\gamma }^{\vee }⟩$. The nonproportional hypothesis excludes $\delta =\pm \gamma$, where zero is not a root and the consecutive-string statement fails.

For ordered simple roots, the Cartan matrix has entries $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$. It determines their angles and relative lengths and therefore determines the corresponding crystallographic root system.

A choice of positive roots selects exactly one of $\alpha$ and $-\alpha$ and is closed under addition whenever the sum is a root.

The simple roots are the positive roots that cannot be expressed as sums of two positive roots. Every root is an integer combination of simple roots with coefficients of one sign.

If a positive root $\alpha$ is not simple, then $\alpha -\beta$ is a positive root for some simple root $\beta$. Iteration writes every positive root as a sum of simple roots whose successive partial sums are roots.

For a simple root $\alpha$, the reflection $s_{\alpha }$ sends $\alpha$ to $-\alpha$ and permutes all other positive roots.

The highest root is the maximal positive root in the root order determined by the simple roots.

The Weyl vector is $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$. It satisfies $⟨\rho ,{\alpha }_i^{\vee }⟩=1$ for every simple root.

The coroot associated with a nonzero root is ${\alpha }^{\vee }=2\alpha /(\alpha ,\alpha )$.

For chosen simple roots, the corresponding coroots are the simple coroots. They form the simple system of the dual root system. The fundamental weights are defined by $⟨{\omega }_j,{\alpha }_i^{\vee }⟩={\delta }_{ij}$, so pairings with simple coroots record coordinates in the fundamental-weight basis.

The dual root system of $\Phi$ is ${\Phi }^{\vee }=\{{\alpha }^{\vee }:\alpha \in \Phi \}$. It has the transposed Cartan matrix, so its Dynkin diagram reverses every multiple-edge arrow; in particular, $B_n$ and $C_n$ are dual.

The root lattice is the integer span of the roots, equivalently of the simple roots.

The Kostant partition function $P(\nu )$ counts the ways to express a weight $\nu$ as a nonnegative integer combination of positive roots. By the Poincaré-Birkhoff-Witt theorem, $P(\lambda -\mu )$ is the multiplicity of the weight $\mu$ in the Verma module $M(\lambda )$.

The Kostant partition function counts nonnegative integer combinations of the positive roots of a root system, without ordering the summands. Multiply the formal geometric series ${\sum }_{k\ge 0}{e}^{-k\alpha }$ for every positive root. The coefficient of $e^{-\nu }$ counts precisely these combinations. Every coefficient is finite because a linear functional strictly positive on the positive roots bounds every participating integer coefficient. The sign of each exponent in the cancelling product must agree with that of this series.

For weights $\mu ,\lambda$, one writes $\mu ⪯\lambda$ when $\lambda -\mu$ is a nonnegative integer combination of the simple roots.

The weight lattice is the set of vectors $\lambda$ for which $⟨\lambda ,{\alpha }^{\vee }⟩$ is an integer for every root $\alpha$.

Restrict a finite-dimensional representation of a complex semisimple Lie algebra to the sl2 subalgebra associated with a root $\alpha$. Its Cartan element is the coroot $h_{\alpha }$. The classification of finite-dimensional sl2 representations gives only integer eigenvalues for this element, so every weight of a representation has integral value on every coroot. This proves containment of all such representation weights in the weight lattice; it does not hold for unrestricted infinite-dimensional modules.

The group ring of a weight lattice $P$ consists of finite sums ${\sum }_{\lambda }{a}_{\lambda }e(\lambda )$ with integer coefficients, where $e(\lambda )e(\mu )=e(\lambda +\mu )$. A finite-dimensional Lie algebra representation has formal character ${\sum }_{\lambda }(\dim V_{\lambda })e(\lambda )$ in this ring. The Weyl group acts by $e(\lambda )\mapsto e(w\lambda )$. The ambient ring need not consist of genuine characters; the representation ring of a semisimple Lie algebra of finite-dimensional modules for a complex semisimple Lie algebra identifies with the invariant subring $\mathbb{Z}[P{]}^W$.

// Target: semisimple-lie-algebra.bigb

For a root reflection on the weight lattice, $s_{\alpha }\lambda =\lambda -m\alpha$ with $m=\lambda (H_{\alpha })\in \mathbb{Z}$. A finite anti-invariant element of the group ring of a weight lattice is a sum of terms $a_{\lambda }[e(\lambda )-e(s_{\alpha }\lambda )]$; fixed weights have zero coefficient. Each pair is divisible by $1-e(\alpha )$, because $1-t^m$ is divisible by $1-t$ in $\mathbb{Z}[t,t^{-1}]$ for every integer $m$. Thus the quotient has finite support.

The formal expression $({\sum }_{n\ge 0}e(n\alpha ))u$ is interpreted coefficientwise. Along each coset of $\mathbb{Z}\alpha$, only finitely many coefficients of $u$ are nonzero and their total is zero by reflection anti-invariance. Partial coefficient sums therefore vanish before and after that finite interval. The result has finite support, even though the individual summands $e(n\alpha )u$ do not eventually vanish.

// Target: lie-algebra.bigb

The Dynkin labels of a weight $\lambda$ are its simple-coroot coordinates ${\lambda }^i=⟨\lambda ,{\alpha }_i^{\vee }⟩$. Thus $\lambda ={\sum }_i{\lambda }^i{\omega }_i$ in the basis of fundamental weights.

A weight is dominant when all its Dynkin labels are nonnegative.

Dominance is a nonnegativity condition on a Weight; integrality additionally requires lattice membership.

An integral weight $\lambda$ is dominant when $⟨\lambda ,{\alpha }_i^{\vee }⟩\ge 0$ for every simple coroot. Equivalently, it is a nonnegative integer combination of the fundamental weights.

This is the lattice-restricted dominant condition on a Weight.

The fundamental weights are dual to the simple coroots: $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$.

The fundamental representation associated with a fundamental weight ${\omega }_i$ is the irreducible highest-weight representation $V({\omega }_i)$.

The Dynkin index of a representation $R$ fixes the trace normalization ${\mathrm{tr}}_R(T^a{T}^b)=I(R){\delta }^{ab}$ up to the convention for the generators. It controls matter contributions to gauge beta functions and mixed anomalies.

The cubic anomaly coefficient is defined by the symmetrized trace of three representation matrices. A conjugate representation has the opposite coefficient, while a real or pseudoreal representation has coefficient zero.

The two-index symmetric representation of $SU(N)$ is the symmetric square of its fundamental representation. It has dimension $N(N+1)/2$, Dynkin index $N+2$, and cubic anomaly coefficient $N+4$ when the fundamental index and anomaly are normalized to one.

The two-index antisymmetric representation of $SU(N)$ is the exterior square of its fundamental representation. It has dimension $N(N-1)/2$, Dynkin index $N-2$, and cubic anomaly coefficient $N-4$ in the same normalization.

On ${\bigwedge }^2{\mathbb{C}}^4$, the invariant Hermitian form and volume tensor define the displayed antilinear map with square one, commuting with SU(4). Its fixed space satisfies $T^{12}=\overline{T^{34}}$, $T^{13}=-\overline{T^{24}}$, and $T^{14}=\overline{T^{23}}$. Three complex choices give six real coordinates, and this fixed space has the original exterior square as its complexification. It is therefore a real representation of dimension six, rather than merely the twelve-dimensional realification of an arbitrary six-dimensional complex representation.

The reflection associated with a root is $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$.

The reflection group of a root system $\Phi$ is the subgroup of the orthogonal group generated by the reflections $s_{\alpha }$ for $\alpha \in \Phi$. It permutes the roots and acts freely and transitively on the fundamental systems.

The Weyl group is generated by the reflections in the roots.

The reflection representation is the action of the Weyl group on the real span of its root system, generated by its Weyl reflections. For an irreducible root system, it is an irreducible representation: if $U$ is an invariant subspace, each root either lies in $U^{\perp }$ or, by subtracting a reflected vector of $U$, lies in $U$. These alternatives partition the roots into orthogonal sets. Irreducibility and spanning force $U=0$ or the whole space.

The longest element of a finite Weyl group is the element sending all positive roots of a root system to negative roots. It is the longest element of a finite Coxeter group. For the G2 root system it is $-I$: its twelve roots form two hexagons, and the dihedral reflection group of a root system contains the rotation through $\pi$. It determines the highest weight of a dual representation.

The Coxeter length $\ell (w)$ is the smallest number of simple reflections whose product is $w$. Its parity gives the sign of a Weyl-group element.

This is a particular length function, namely word length for the simple reflections.

If $\lambda$ and $w\lambda$ both lie in the closed dominant chamber, then $w$ is a product of simple reflections whose walls contain $\lambda$. In particular, $w\lambda =\lambda$.

For a chosen positive system of a root system $\Pi$, the inversion set isFor a finite Weyl group, $|N(w)|=\ell (w)$.

Its cardinality is the Coxeter length; this root-sign definition should not be silently identified with an unspecified permutation inversion convention.

In the geometric representation of a Coxeter system,andReplacing “positive” by “negative” reverses either length inequality.

A Dynkin diagram records the angles and relative lengths of the simple roots. A multiple-edge arrow points toward the shorter root.

The connected Dynkin diagrams of finite reduced crystallographic root systems are $A_r$, $B_r$, $C_r$, $D_r$, $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$. Bonds encode products of off-diagonal Cartan integers; multiple-bond arrows point toward shorter roots. The simply laced branching diagrams have arm lengths $(1,1,r-3)$ for $D_r$ and $(1,2,2)$, $(1,2,3)$, $(1,2,4)$ for the three exceptional $E$ types. The $B$ and $C$ chains have an end double bond, $F_4$ a middle double bond, and $G_2$ a triple bond. This edge weighting is distinct from the vertex labels used to classify nilpotent orbits.

The underlying graph of a finite Dynkin diagram is a forest. Normalize every simple root to length one. Each bonded pair has inner product at most $-1/2$, and every other distinct pair has nonpositive inner product. A cycle with $m$ vertices would make the squared norm of their sum at most $m-m=0$, contradicting their linear independence in an inner product space. Multiple bonds denote root-length information, not graph-theoretic parallel-edge cycles.

The finite Dynkin diagram $G_2$ has two vertices joined by a triple bond, with the arrow toward the short root. The squared root-length ratio is three. It represents the G2 root system.

The finite Dynkin diagram $F_4$ is a four-vertex chain whose central bond is double. The two consecutive long roots lie on one side, the two consecutive short roots on the other, and the arrow points toward the short pair. The remaining two bonds are single.

The finite Dynkin diagrams $E_6,E_7,E_8$ are simply laced trees with one trivalent vertex. Their three arm lengths, counting edges, are respectively $(1,2,2)$, $(1,2,3)$ and $(1,2,4)$.

For $n\ge 4$, the finite Dynkin diagram $D_n$ is a simply laced tree with one trivalent vertex and three arms containing $1,1,n-3$ edges. All its roots have the same length.

For $n\ge 2$, the finite Dynkin diagram $C_n$ is a chain with a terminal double bond. Its arrow points from the terminal long root toward the adjacent short root. It represents the Cn root system; $C_2$ is isomorphic to $B_2$ after relabelling.

For $n\ge 1$, the finite Dynkin diagram $A_n$ is a chain of $n$ vertices joined by single bonds. It represents the root system of ${\mathfrak{sl}}_{n+1}$.

An irreducible root system is simply laced when all roots have the same length, equivalently when its Dynkin diagram has only single edges. This is also equivalent towhenever $\alpha \ne \pm \beta$.

The connected positive-definite simply-laced Dynkin diagrams are $A_n$ for $n\ge 1$, $D_n$ for $n\ge 4$, and the exceptional diagrams $E_6,E_7,E_8$.

The $D_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$. The simple rootsform the three-node chain ${\alpha }_2-{\alpha }_1-{\alpha }_3$, so $D_3\cong A_3$.

The root-system isomorphism $D_3\cong A_3$ and the classification of complex simple Lie algebras give ${\mathfrak{so}}_6(\mathbb{C})\cong {\mathfrak{sl}}_4(\mathbb{C})$.

An extended, or affine, Dynkin diagram adjoins the root ${\alpha }_0=-\theta$, where $\theta$ is the highest root.

The roots of maximal length in a root system form a root subsystem. A chamber for the original system lies in a unique chamber for this long-root subsystem, because its interior avoids every long-root reflecting hyperplane.

The $G_2$ root system has six short and six long roots. For a short simple root ${\alpha }_1$ and a long simple root ${\alpha }_2$, its positive roots are

These are the root data associated with G2.

If two simple roots have angle $5\pi /6$, normalize them to unit vectors $u,v$ with $(u,v)=-\sqrt{3}/2$. For any other unit simple root $w$, set $r=(u,w)\le 0$, $s=(v,w)\le 0$. Its three-root Gram matrix has determinant $1/4-r^2-s^2-\sqrt{3}rs$. Any nonzero $r$ or $s$ has absolute value at least $1/2$, contradicting positive definiteness. Thus all other simple roots are orthogonal to $u,v$. An irreducible root system containing this triple bond must be the rank-two G2 root system.

For a short root $\alpha$ in the G2 root system, restriction of the Adjoint representation to the sl2 subalgebra associated with a root givesHere $V(n)$ is the irreducible sl2 Lie algebra representation of highest weight $n$ and dimension $n+1$. Every nonzero root vector $x\in {\mathfrak{g}}_{\alpha }$ has $\dim Z_{\mathfrak{g}}(x)=6$: the raising operator has a one-dimensional kernel in each of these six irreducible summands.

For short and long simple roots $\alpha ,\beta$ of $G_2$, the long roots form an $A_2$ root system with simple roots ${\delta }_1=\beta$ and ${\delta }_2=3\alpha +\beta$. If ${\omega }_1,{\omega }_2$ are the corresponding $A_2$ fundamental weights, then $\alpha ={\omega }_2-{\omega }_1$ and ${\alpha }^{\vee }={\delta }_2^{\vee }-{\delta }_1^{\vee }$.

The seven-dimensional fundamental $G_2$ representation has weights zero and the six short roots. Under its long-root $A_2$ subalgebra, these are the weights of the defining three-dimensional representation, its dual, and the trivial representation:

For roots $\alpha \ne \pm \beta$,where $\theta$ is their angle. Integrality gives an integer, and nonparallelism makes it strictly smaller than four.

Every reduced crystallographic rank-two root system is isomorphic to $A_1\times A_1$, $A_2$, $B_2$, or $G_2$. Their simple-root angles are respectively $\pi /2$, $2\pi /3$, $3\pi /4$, and $5\pi /6$.

The two simple reflections generate a dihedral group. For $A_1\times A_1$, $A_2$, $B_2$, and $G_2$, the rotation has order $2,3,4,6$, so the Weyl groups have orders $4,6,8,12$ respectively.

A fundamental system $\Delta$ is a basis made of roots such that every root is a linear combination of $\Delta$ whose nonzero coefficients all have the same sign. Its members are the simple roots.

The positive system associated with a fundamental system of a root system $\Delta$ consists of the roots whose coordinates in the basis $\Delta$ are nonnegative.