Root subsystem - Knowlpedia

Definition

Let $\Phi$ be a root system in a Euclidean space $V$ . A subset $\Psi\subseteq\Phi$ is a root subsystem if $\Psi$ is itself a root system in $\operatorname{span}_{\mathbb R}(\Psi)$ with the induced inner product. Equivalently, $\Psi$ spans its ambient subspace and

s_\alpha(\Psi)=\Psi \qquad\text{for every }\alpha\in\Psi,

where $s_\alpha$ is reflection in the hyperplane perpendicular to $\alpha$ .

Full and closed subsystems

A subsystem is full in its span if

\Psi=\Phi\cap\operatorname{span}_{\mathbb R}(\Psi).

It is closed in $\Phi$ if $\alpha,\beta\in\Psi$ and $\alpha+\beta\in\Phi$ imply $\alpha+\beta\in\Psi$ . These adjectives express additional conditions: “root subsystem” alone need not mean full or closed in every source.

If $I$ is a subset of a chosen base $\Delta$ of simple roots, then

\Phi_I:=\Phi\cap\operatorname{span}_{\mathbb Z}(I) =\Phi\cap\operatorname{span}_{\mathbb R}(I)

is a full closed subsystem. Its Dynkin diagram is obtained by retaining the vertices in $I$ and the edges between them.

Lie-algebra interpretation

For a complex semisimple Lie algebra with root-space decomposition

\mathfrak g=\mathfrak h\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha,

closed subsystems select collections of root spaces that can participate in regular subalgebras. The subsystem $\Phi_I$ gives the semisimple part of the Levi subalgebra attached to $I$ .

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Springer, 2002, Chapter VI, §1. Publisher record.
Toshio Oshima, “A classification of subsystems of a root system,” 2006, §§1–2. arXiv record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §§2, 5. arXiv record.