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Root and root space

Definition

Let $g$ be a finite-dimensional complex semisimple Lie algebra and let $h$ be a Cartan subalgebra of $g$ (Cartan subalgebra). For $α \in h^{*}$ define the root space

$g_{α} = {x \in g : [H, x] = α (H) x for all H \in h},$

using $[H, x] = {ad}_{H} (x)$ from Derivations of Lie algebras.

A root of $g$ with respect to $h$ is a nonzero functional $α \in h^{*}$ with $g_{α} \neq 0$ . The set of roots is written $Φ (g, h)$ , or simply $Φ$ . By convention, $g_{λ} = 0$ when a functional $λ \in h^{*}$ is neither zero nor a root.

Depends on

Used by

Cartan integers are integers Corollary
Opposite root spaces pair nondegenerately Corollary
Regular elements form a dense Zariski-open subset of a Cartan subalgebra Corollary
The only scalar multiples of a root that are roots are plus or minus the root Corollary
Cayley transform of a theta-stable Cartan subalgebra Definition
Complex semisimple algebraic group, Borel, and flag variety Definition
Killing-dual vector of a root Definition
Regular root hyperplanes Definition
Root reflection defined by a coroot Definition
Cartan subalgebra and roots of sl₂ Example
Diagonal Cartan subalgebra and roots of slₙ Example
Root strings in type A₂ Example
Root systems B₂ and C₂ from matrix Lie algebras Example
Root-space brackets for matrix units Example
The Weyl reflection in sl₂ Example
All integer multiples of a root are roots False statement
If alpha and beta are roots then alpha plus beta is always a root False statement
Root spaces can have arbitrary dimension in a complex semisimple Lie algebra False statement
Algebraic root subgroups from root exponentials Lemma
Chevalley basis and real structure constants Lemma
Flag line-bundle degree on a minimal-parabolic fiber Lemma
Minimal parabolic from one negative simple root Lemma
Rational highest-weight modules from adjoint Plücker vectors Lemma
The central reduction of U(sl2) is simple away from the finite-dimensional central characters Lemma
The Killing length of a root is nonzero Lemma
Brackets of root spaces Proposition
Centralizer dimension from vanishing roots Proposition
Dimension formula from roots Proposition
Orthogonality of root spaces and nondegeneracy on the Cartan subalgebra Proposition
Root reflections are induced by inner automorphisms Proposition
Root vectors shift weights Proposition
The bracket of opposite root spaces is the root line Proposition
The center is the common kernel of the roots inside the Cartan subalgebra Proposition
The roots form a reduced crystallographic Euclidean root system Proposition
Existence and uniqueness up to isomorphism of the split real form Theorem
Generic irreducibility and the exceptional parameter lattice Theorem
Root reflections preserve the root set Theorem
Root spaces of a complex semisimple Lie algebra are one-dimensional Theorem
Root-space decomposition Theorem
Roots of a complex semisimple Lie algebra form a reduced crystallographic root system Theorem

…and 2 more results.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19–24 (standard reference, not scraped)