Editing Semisimple Lie algebra (section)

== Real semisimple Lie algebra ==
For a semisimple Lie algebra over a field that has characteristic zero but is not algebraically closed, there is no general structure theory like the one for those over an algebraically closed field of characteristic zero. But over the field of real numbers, there are still the structure results.

Let <math>\mathfrak g</math> be a finite-dimensional real semisimple Lie algebra and <math>\mathfrak{g}^{\mathbb{C}} = \mathfrak{g} \otimes_{\mathbb{R}} \mathbb{C}</math> the complexification of it (which is again semisimple). The real Lie algebra <math>\mathfrak g</math> is called a [[real form]] of <math>\mathfrak{g}^{\mathbb{C}}</math>. A real form is called a compact form if the Killing form on it is negative-definite; it is necessarily the Lie algebra of a compact Lie group (hence, the name).

=== Compact case ===
Suppose <math>\mathfrak g</math> is a compact form and <math>\mathfrak h \subset \mathfrak g</math> a maximal abelian subspace. One can show (for example, from the fact <math>\mathfrak g</math> is the Lie algebra of a compact Lie group) that <math>\operatorname{ad}(\mathfrak h)</math> consists of skew-Hermitian matrices, diagonalizable over <math>\mathbb{C}</math> with imaginary eigenvalues. Hence, <math>\mathfrak h^{\mathbb{C}}</math> is a [[Cartan subalgebra]] of <math>\mathfrak{g}^{\mathbb{C}}</math> and there results in the root space decomposition (cf. [[#Structure]])
:<math>\mathfrak{g}^{\mathbb{C}} = \mathfrak{h}^{\mathbb{C}} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}</math>
where each <math>\alpha \in \Phi</math> is real-valued on <math>i \mathfrak{h}</math>; thus, can be identified with a real-linear functional on the real vector space <math>i \mathfrak{h}</math>.

For example, let <math>\mathfrak{g} = \mathfrak{su}(n)</math> and take <math>\mathfrak h \subset \mathfrak g</math> the subspace of all diagonal matrices. Note <math>\mathfrak{g}^{\mathbb{C}} = \mathfrak{sl}_n \mathbb{C}</math>. Let <math>e_i</math> be the linear functional on <math>\mathfrak{h}^{\mathbb{C}}</math> given by <math>e_i(H) = h_i</math> for <math>H = \operatorname{diag}(h_1, \dots, h_n)</math>. Then for each <math>H \in \mathfrak{h}^{\mathbb{C}}</math>,
:<math>[H, E_{ij}] = (e_i(H) - e_j(H)) E_{ij}</math>
where <math>E_{ij}</math> is the matrix that has 1 on the <math>(i, j)</math>-th spot and zero elsewhere. Hence, each root <math>\alpha</math> is of the form <math>\alpha = e_i - e_j, i \ne j</math> and the root space decomposition is the decomposition of matrices:<ref>{{harvnb|Knapp|2002|loc=Ch. IV, § 1, Example 1.}}</ref>
:<math>\mathfrak{g}^{\mathbb{C}} = \mathfrak{h}^{\mathbb{C}} \oplus \bigoplus_{i \ne j} \mathbb{C} E_{ij}.</math>

=== Noncompact case ===
Suppose <math>\mathfrak g</math> is not necessarily a compact form (i.e., the signature of the Killing form is not all negative). Suppose, moreover, it has a [[Cartan involution]] <math>\theta</math> and let <math>\mathfrak g = \mathfrak k \oplus \mathfrak p</math> be the eigenspace decomposition of <math>\theta</math>, where <math>\mathfrak k, \mathfrak p</math> are the eigenspaces for 1 and -1, respectively. For example, if <math>\mathfrak g = \mathfrak{sl}_n \mathbb{R}</math> and <math>\theta</math> the negative transpose, then <math>\mathfrak k = \mathfrak{so}(n)</math>.

Let <math>\mathfrak a \subset \mathfrak p</math> be a maximal abelian subspace. Now, <math>\operatorname{ad}(\mathfrak p)</math> consists of symmetric matrices (with respect to a suitable inner product) and thus the operators in <math>\operatorname{ad}(\mathfrak a)</math> are simultaneously diagonalizable, with real eigenvalues. By repeating the arguments for the algebraically closed base field, one obtains the decomposition (called the '''restricted root space decomposition'''):<ref>{{harvnb|Knapp|2002|loc=Ch. V, § 2, Proposition 5.9.}}</ref>
:<math>\mathfrak g = \mathfrak g_0 \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}</math>
where
*the elements in <math>\Phi</math> are called the [[restricted root]]s,
*<math>\theta(\mathfrak{g}_{\alpha}) = \mathfrak{g}_{-\alpha}</math> for any linear functional <math>\alpha</math>; in particular, <math>-\Phi \subset \Phi</math>,
*<math>\mathfrak g_0 = \mathfrak a \oplus Z_{\mathfrak k}(\mathfrak a)</math>.
Moreover, <math>\Phi</math> is a [[root system]] but not necessarily reduced one (i.e., it can happen <math>\alpha, 2\alpha</math> are both roots).