Levi subalgebra - Knowlpedia

Definition

Let $\mathfrak g$ be a complex semisimple Lie algebra, choose a Cartan subalgebra $\mathfrak h$ , a root system $\Phi$ , and a base $\Delta$ of simple roots. For $I\subseteq\Delta$ , put

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

The associated standard Levi subalgebra is

\mathfrak l_I =\mathfrak h\oplus\bigoplus_{\alpha\in\Phi_I}\mathfrak g_\alpha.

A Levi subalgebra of $\mathfrak g$ is a subalgebra conjugate under an inner automorphism to some $\mathfrak l_I$ .

Structure

The Lie algebra $\mathfrak l_I$ is reductive. Its derived algebra and center are

[\mathfrak l_I,\mathfrak l_I] =\mathfrak h_I\oplus\bigoplus_{\alpha\in\Phi_I}\mathfrak g_\alpha, \qquad Z(\mathfrak l_I) =\{H\in\mathfrak h:\alpha(H)=0\text{ for every }\alpha\in I\},

where $\mathfrak h_I$ is the span of the coroots belonging to $I$ . Thus

\mathfrak l_I=Z(\mathfrak l_I)\oplus[\mathfrak l_I,\mathfrak l_I],

and the semisimple summand has root system $\Phi_I$ .

Relation to the Dynkin diagram

The Dynkin diagram of $[\mathfrak l_I,\mathfrak l_I]$ is the subdiagram induced by the vertices in $I$ . The entire $\mathfrak l_I$ , however, retains the full Cartan subalgebra $\mathfrak h$ , so every Levi subalgebra has the same rank as $\mathfrak g$ .

Levi subalgebras are precisely the reductive factors of parabolic subalgebras. This use of “Levi” is related to, but more specific than, a Levi factor in the Levi decomposition of an arbitrary finite-dimensional Lie algebra.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapter II. Publisher record.
Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 7–9, Springer, 2005, Chapter VIII, §3. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §2. arXiv record.