Maximal Levi subalgebra

Definition

A maximal Levi subalgebra of a complex semisimple Lie algebra $\mathfrak g$ is a proper Levi subalgebra that is maximal, under inclusion, among proper Levi subalgebras of $\mathfrak g$ .

After choosing simple roots $\Delta$ , every standard maximal Levi subalgebra has the form

\mathfrak l_{\Delta\setminus\{\alpha_0\}} =\mathfrak h\oplus \bigoplus_{\beta\in\Phi\cap \operatorname{span}_{\mathbb Z}(\Delta\setminus\{\alpha_0\})} \mathfrak g_\beta

for one $\alpha_0\in\Delta$ . Thus it is obtained by removing one simple root from the semisimple root data while retaining the full Cartan subalgebra $\mathfrak h$ .

Structure

If $\mathfrak g$ is semisimple, the derived algebra of this maximal Levi has the Dynkin diagram obtained by deleting the vertex $\alpha_0$ and all incident edges. Its center is one-dimensional:

\dim Z\!\left(\mathfrak l_{\Delta\setminus\{\alpha_0\}}\right)=1.

The Levi subalgebra itself has maximal rank in $\mathfrak g$ .

Important distinction

“Maximal Levi” does not ordinarily mean maximal among all Lie subalgebras. A proper Levi subalgebra lies inside the corresponding proper parabolic subalgebra, so there is usually a larger proper subalgebra between it and $\mathfrak g$ .

For example, deleting a suitable node from the $A_4$ diagram gives a semisimple part of type $A_2\sqcup A_1$ ; retaining the Cartan contributes a one-dimensional center, producing a reductive algebra isomorphic to

\mathfrak{sl}_3\oplus\mathfrak{sl}_2\oplus\mathbb C.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapter II. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §2. arXiv record.