Compact real form - Knowlpedia

Definition

Let $\mathfrak g$ be a finite-dimensional complex semisimple Lie algebra. A real Lie subalgebra $\mathfrak u\subset\mathfrak g$ is a compact real form of $\mathfrak g$ if

\mathfrak u\otimes_{\mathbb R}\mathbb C\cong\mathfrak g

through the complex-linear extension of the inclusion, and the Killing form of $\mathfrak u$ is negative definite.

Equivalently, $\mathfrak u$ is a real form of $\mathfrak g$ that is the Lie algebra of a compact semisimple Lie group.

Existence and uniqueness

Every complex semisimple Lie algebra has a compact real form. Any two compact real forms are conjugate by an inner automorphism of the complex Lie algebra.

One construction starts from a Chevalley basis $h_i,e_\alpha$ and takes the real span of

i h_i,\qquad e_\alpha-e_{-\alpha},\qquad i(e_\alpha+e_{-\alpha})

for the simple coroot directions and positive roots $\alpha$ . The resulting real algebra has negative-definite Killing form.

Classical and exceptional examples

The standard compact real forms include

\mathfrak{su}(n)\subset\mathfrak{sl}_n(\mathbb C), \qquad \mathfrak{so}(n)\subset\mathfrak{so}_n(\mathbb C), \qquad \mathfrak{sp}(n)\subset\mathfrak{sp}_{2n}(\mathbb C).

The compact real form of $\mathfrak e_7$ is often denoted $\mathfrak e_{7(-133)}$ , where $-133$ is the signature of its negative-definite Killing form.

Compact conjugation

Complex conjugation with respect to the real vector space $\mathfrak u$ is an antilinear involutive Lie algebra automorphism

\sigma:\mathfrak g\to\mathfrak g, \qquad \mathfrak g^\sigma=\mathfrak u.

Conversely, an antilinear involution whose fixed algebra has negative-definite Killing form determines a compact real form. This turns statements about complex roots and representations into statements about compact Lie groups after imposing the appropriate reality condition.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapters II, IV, VI. Publisher record.
Jean-Pierre Serre, Complex Semisimple Lie Algebras, Springer, 1987, Chapter V. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §1. arXiv record.