Editing Semisimple Lie algebra (section)

==Representation theory of semisimple Lie algebras ==
{{main|Representation theory of semisimple Lie algebras}}

Let <math>\mathfrak g</math> be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. Then, as in [[#Structure]], <math display="inline">\mathfrak g = \mathfrak h \oplus \bigoplus_{\alpha \in \Phi} \mathfrak g_{\alpha}</math> where <math>\Phi</math> is the root system. Choose the simple roots in <math>\Phi</math>; a root <math>\alpha</math> of <math>\Phi</math> is then called [[positive root|positive]] and is denoted by <math>\alpha > 0</math> if it is a linear combination of the simple roots with non-negative integer coefficients. Let <math display="inline">\mathfrak b = \mathfrak h \oplus \bigoplus_{\alpha > 0} \mathfrak g_{\alpha}</math>, which is a maximal solvable subalgebra of <math>\mathfrak g</math>, the [[Borel subalgebra]].

Let ''V'' be a (possibly-infinite-dimensional) simple <math>\mathfrak g</math>-module. If ''V'' happens to admit a <math>\mathfrak b</math>-weight vector <math>v_0</math>,<ref>A <math>\mathfrak{b}</math>-weight vector is also called a [[primitive element (Lie algebra)|primitive element]], especially in older textbooks.</ref> then it is unique up to scaling and is called the [[highest weight vector]] of ''V''. It is also an <math>\mathfrak h</math>-weight vector and the <math>\mathfrak h</math>-weight of <math>v_0</math>, a linear functional of <math>\mathfrak h</math>, is called the [[highest weight]] of ''V''. The basic yet nontrivial facts<ref>In textbooks, these facts is usually established by the theory of [[Verma module]]s.</ref> then are (1) to each linear functional <math>\mu \in \mathfrak h^*</math>, there exists a simple <math>\mathfrak g</math>-module <math>V^{\mu}</math> having <math>\mu</math> as its highest weight and (2) two simple modules having the same highest weight are equivalent. In short, there exists a bijection between <math>\mathfrak h^*</math> and the set of the equivalence classes of simple <math>\mathfrak g</math>-modules admitting a Borel-weight vector.

For applications, one is often interested in a finite-dimensional simple <math>\mathfrak g</math>-module (a finite-dimensional irreducible representation). This is especially the case when <math>\mathfrak g</math> is the Lie algebra of a [[Lie group]] (or complexification of such), since, via the [[Lie correspondence]], a Lie algebra representation can be integrated to a Lie group representation when the obstructions are overcome. The next criterion then addresses this need: by the [[positive Weyl chamber]] <math>C \subset \mathfrak{h}^*</math>, we mean the convex cone <math>C = \{ \mu \in \mathfrak{h}^* | \mu(h_{\alpha}) \ge 0, \alpha \in \Phi > 0 \}</math> where <math>h_{\alpha} \in [\mathfrak g_{\alpha}, \mathfrak g_{-\alpha}]</math> is a unique vector such that <math>\alpha(h_{\alpha}) = 2</math>. The criterion then reads:<ref>{{harvnb|Serre|2000|loc=Ch. VII, § 4, Theorem 3.}}</ref>
*<math>\dim V^{\mu} < \infty</math> if and only if, for each positive root <math>\alpha > 0</math>, (1) <math>\mu(h_{\alpha})</math> is an integer and (2) <math>\mu</math> lies in <math>C</math>.
A linear functional <math>\mu</math> satisfying the above equivalent condition is called a dominant integral weight. Hence, in summary, there exists a bijection between the dominant integral weights and the equivalence classes of finite-dimensional simple <math>\mathfrak g</math>-modules, the result known as the [[theorem of the highest weight]]. The character of a finite-dimensional simple module in turns is computed by the [[Weyl character formula]].<!-- expand this -->

The [[Weyl's completely reducibility theorem|theorem due to Weyl]] says that, over a field of characteristic zero, every finite-dimensional [[representation of a Lie algebra|module]] of a semisimple Lie algebra <math>\mathfrak g</math> is [[semisimple representation|completely reducible]]; i.e., it is a direct sum of simple <math>\mathfrak g</math>-modules. Hence, the above results then apply to finite-dimensional representations of a semisimple Lie algebra.