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

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.

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.

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 ${\alpha }^{\vee }=2\alpha /(\alpha ,\alpha )$.

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

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 $\alpha$.

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.

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 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 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.

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, $|N(w)|=\ell (w)$.

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.

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

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

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.

The fundamental chamber is the connected componentof the complement of the reflecting hyperplanes. The Weyl group acts freely and transitively on its chambers.

For a root basis $\Delta$, the closed dominant chamber isEvery Weyl-group orbit meets it in exactly one point.