Editing Semisimple Lie algebra (section)

== Structure ==
Let <math>\mathfrak g</math> be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. The structure of <math>\mathfrak g</math> can be described by an [[adjoint action]] of a certain distinguished subalgebra on it, a [[Cartan subalgebra]]. By definition,<ref>This is a definition of a Cartan subalgebra of a semisimple Lie algebra and coincides with the general one.</ref> a [[Cartan subalgebra]] (also called a maximal [[toral subalgebra]]) <math>\mathfrak h</math> of <math>\mathfrak g</math> is a maximal subalgebra such that, for each <math>h \in \mathfrak h</math>, <math>\operatorname{ad}(h)</math> is [[diagonalizable matrix|diagonalizable]]. As it turns out, <math>\mathfrak h</math> is abelian and so all the operators in <math>\operatorname{ad}(\mathfrak h)</math> are [[simultaneously diagonalizable]]. For each linear functional <math>\alpha</math> of <math>\mathfrak h</math>, let
:<math>\mathfrak{g}_{\alpha} = \{ x \in \mathfrak{g} | \operatorname{ad}(h) x := [h, x] = \alpha(h) x \, \text{ for all } h \in \mathfrak h \}</math>.
(Note that <math>\mathfrak{g}_0</math> is the [[centralizer (Lie algebra)|centralizer]] of <math>\mathfrak h</math>.) Then

{{math_theorem
| name = Root space decomposition
| math_statement = <ref>{{harvnb|Serre|2000|loc=Ch. VI, § 1.}}</ref> Given a Cartan subalgebra <math>\mathfrak{h}</math>, it holds that <math>\mathfrak{g}_0 = \mathfrak{h}</math> and there is a decomposition (as an <math>\mathfrak h</math>-module):
:<math>\mathfrak g = \mathfrak h \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}</math>
where <math>\Phi</math> is the set of all nonzero linear functionals <math>\alpha</math> of <math>\mathfrak h</math> such that <math>\mathfrak g_{\alpha} \ne \{0\}</math>. Moreover, for each <math>\alpha, \beta \in \Phi</math>,
*<math>[\mathfrak g_{\alpha}, \mathfrak g_{\beta}] \subseteq \mathfrak g_{\alpha + \beta}</math>, which is the equality if <math>\alpha + \beta \ne 0</math>.
*<math>[\mathfrak{g}_{\alpha}, \mathfrak{g}_{-\alpha}] \oplus \mathfrak{g}_{-\alpha} \oplus \mathfrak{g}_{\alpha} \simeq \mathfrak{sl}_2</math> as a Lie algebra.
*<math>\dim \mathfrak g_{\alpha} = 1</math>; in particular, <math>\dim \mathfrak g = \dim \mathfrak h + \# \Phi</math>.
*<math>\mathfrak g_{2\alpha} = \{0\}</math>; in other words, <math>2 \alpha \not\in \Phi</math>.
*With respect to the Killing form ''B'', <math>\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}</math> are orthogonal to each other if <math>\alpha + \beta \ne 0</math>; the restriction of ''B'' to <math>\mathfrak h</math> is nondegenerate.
}}
(The most difficult item to show is <math>\dim \mathfrak{g}_{\alpha} = 1</math>. The standard proofs all use some facts in the [[representation theory of sl 2|representation theory of <math>\mathfrak{sl}_2</math>]]; e.g., Serre uses the fact that an <math>\mathfrak{sl}_2</math>-module with a primitive element of negative weight is infinite-dimensional, contradicting <math>\dim \mathfrak g < \infty</math>.)

Let <math>h_{\alpha} \in \mathfrak{h}, e_{\alpha} \in \mathfrak{g}_{\alpha}, f_{\alpha} \in \mathfrak{g}_{-\alpha}</math> with the commutation relations <math>[e_{\alpha}, f_{\alpha}] = h_{\alpha}, [h_{\alpha}, e_{\alpha}] = 2e_{\alpha}, [h_{\alpha}, f_{\alpha}] = -2f_{\alpha}</math>; i.e., the <math>h_{\alpha}, e_{\alpha}, f_{\alpha}</math> correspond to the standard basis of <math>\mathfrak{sl}_2</math>.

The linear functionals in <math>\Phi</math> are called the '''roots''' of <math>\mathfrak g</math> relative to <math>\mathfrak h</math>. The roots span <math>\mathfrak h^*</math> (since if <math>\alpha(h) = 0, \alpha \in \Phi</math>, then <math>\operatorname{ad}(h)</math> is the zero operator; i.e., <math>h</math> is in the center, which is zero.) Moreover, from the representation theory of <math>\mathfrak{sl}_2</math>, one deduces the following symmetry and integral properties of <math>\Phi</math>: for each <math>\alpha, \beta \in \Phi</math>,
{{bulleted list
|The endomorphism
:<math>s_{\alpha} : \mathfrak{h}^* \to \mathfrak{h}^*, \, \gamma \mapsto \gamma - \gamma(h_{\alpha}) \alpha</math>
leaves <math>\Phi</math> invariant (i.e., <math>s_{\alpha}(\Phi) \subset \Phi</math>).
|<math>\beta(h_{\alpha})</math> is an integer.}}
Note that <math>s_{\alpha}</math> has the properties (1) <math>s_{\alpha}(\alpha) = -\alpha</math> and (2) the fixed-point set is <math>\{ \gamma \in \mathfrak{h}^* | \gamma(h_\alpha) = 0 \}</math>, which means that <math>s_{\alpha}</math> is the reflection with respect to the hyperplane corresponding to <math>\alpha</math>. The above then says that <math>\Phi</math> is a [[root system]].

It follows from the general theory of a root system that <math>\Phi</math> contains a basis <math>\alpha_1, \dots, \alpha_l</math> of <math>\mathfrak{h}^*</math> such that each root is a linear combination of <math>\alpha_1, \dots, \alpha_l</math> with integer coefficients of the same sign; the roots <math>\alpha_i</math> are called [[Root system#Positive roots and simple roots|simple roots]]. Let <math>e_i = e_{\alpha_i}</math>, etc. Then the <math>3l</math> elements <math>e_i, f_i, h_i</math> (called '''Chevalley generators''') generate <math>\mathfrak g</math> as a Lie algebra. Moreover, they satisfy the relations (called '''Serre relations'''): with <math>a_{ij} = \alpha_j(h_i)</math>,
:<math>[h_i, h_j] = 0,</math>
:<math>[e_i, f_i] = h_i, [e_i, f_j] = 0, i \ne j,</math>
:<math>[h_i, e_j] = a_{ij} e_j, [h_i, f_j] = -a_{ij} f_j,</math>
:<math>\operatorname{ad}(e_i)^{-a_{ij} + 1}(e_j) = \operatorname{ad}(f_i)^{-a_{ij} + 1}(f_j) = 0, i \ne j</math>.
The converse of this is also true: i.e., the Lie algebra generated by the generators and the relations like the above is a (finite-dimensional) semisimple Lie algebra that has the root space decomposition as above (provided the <math>[a_{ij}]_{1 \le i, j \le l}</math> is a [[Cartan matrix]]). This is a [[Serre's theorem on a semisimple Lie algebra|theorem of Serre]]. In particular, two semisimple Lie algebras are isomorphic if they have the same root system.

The implication of the axiomatic nature of a root system and Serre's theorem is that one can enumerate all possible root systems; hence, "all possible" semisimple Lie algebras (finite-dimensional over an algebraically closed field of characteristic zero).

The '''Weyl group''' is the group of linear transformations of <math>\mathfrak{h}^* \simeq \mathfrak{h}</math> generated by the <math>s_\alpha</math>'s. The Weyl group is an important symmetry of the problem; for example, the weights of any finite-dimensional representation of <math>\mathfrak{g}</math> are invariant under the Weyl group.<ref>{{harvnb|Hall|2015}} Theorem 9.3</ref>