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Coroot of a Lie-algebra root

Definition

Assume the Axiom of Choice. Let $α$ be a root of a finite-dimensional complex semisimple Lie algebra $g$ with respect to a Cartan subalgebra $h$ , and let $H_{α} \in h$ be its Killing-dual vector (Killing-dual vector of a root). By The Killing length of a root is nonzero the number $α (H_{α})$ equals $B (H_{α}, H_{α})$ and is nonzero, so the following element of $h$ is well defined:

$h_{α} = \frac{2 H_{α}}{α (H_{α})} .$

It is called the coroot of $α$ . It satisfies $B (h_{α}, H) = \frac{2 α (H)}{α (H_{α})}$ for all $H \in h$ , and $α (h_{α}) = 2$ .

Depends on

Used by

Cartan integers are integers Corollary
The only scalar multiples of a root that are roots are plus or minus the root Corollary
A nondominant integral highest-weight module can be infinite-dimensional Counterexample
Cayley transform of a theta-stable Cartan subalgebra Definition
Integral, dominant, and strictly dominant weights Definition
Root reflection defined by a coroot Definition
Root strings in type A₂ Example
Standard and dual representations of slₙ Example
The Killing form identifies roots with coroot directions Example
The root sl₂ triple inside slₙ Example
The Weyl reflection in sl₂ Example
Dominance depends on a positive system False statement
Finite-dimensionality requires dominance integrality False statement
Root spaces can have arbitrary dimension in a complex semisimple Lie algebra False statement
Chevalley basis and real structure constants Lemma
Simple reflections preserve weight multiplicities Lemma
Root reflections are induced by inner automorphisms Proposition
The roots form a reduced crystallographic Euclidean root system Proposition
Cayley transforms connect theta-stable Cartans in the classification Theorem
Classification of real forms by Vogan diagrams Theorem
Compact roots form a reduced crystallographic root system Theorem
Existence and uniqueness up to isomorphism of the split real form Theorem
Root reflections preserve the root set Theorem
Root spaces of a complex semisimple Lie algebra are one-dimensional Theorem
Roots of a complex semisimple Lie algebra form a reduced crystallographic root system Theorem
The root sl₂ triple Theorem
The root-string property Theorem

Dependency tree · two levels

17 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)