Editing Semisimple Lie algebra (section)

== The case of sl(n,C)==
If <math>\mathfrak{g}=\mathrm{sl}(n,\mathbb{C})</math>, then <math>\mathfrak{h}</math> may be taken to be the diagonal subalgebra of <math>\mathfrak{g}</math>, consisting of diagonal matrices whose diagonal entries sum to zero. Since <math>\mathfrak{h}</math> has dimension <math>n-1</math>, we see that <math>\mathrm{sl}(n;\mathbb{C})</math> has rank <math>n-1</math>.

The root vectors <math>X</math> in this case may be taken to be the matrices <math>E_{i,j}</math> with <math>i\neq j</math>, where <math>E_{i,j}</math> is the matrix with a 1 in the <math>(i,j)</math> spot and zeros elsewhere.<ref>{{harvnb|Hall|2015}} Section 7.7.1</ref> If <math>H</math> is a diagonal matrix with diagonal entries <math>\lambda_1,\ldots,\lambda_n</math>, then we have
:<math>[H,E_{i,j}]=(\lambda_i-\lambda_j)E_{i,j}</math>.
Thus, the roots for <math>\mathrm{sl}(n,\mathbb{C})</math> are the linear functionals <math>\alpha_{i,j}</math> given by
:<math>\alpha_{i,j}(H)=\lambda_i-\lambda_j</math>.
After identifying <math>\mathfrak{h}</math> with its dual, the roots become the vectors <math>\alpha_{i,j}:=e_i-e_j</math> in the space of <math>n</math>-tuples that sum to zero. This is the root system [[Root system#An|known as <math>A_{n-1}</math>]] in the conventional labeling.

The reflection associated to the root <math>\alpha_{i,j}</math> acts on <math>\mathfrak{h}</math> by transposing the <math>i</math> and <math>j</math> diagonal entries. The Weyl group is then just the permutation group on <math>n</math> elements, acting by permuting the diagonal entries of matrices in <math>\mathfrak{h}</math>.<!-- This is the old version of the structure section; it may (or may not) be needed to incorporated into the article
== Structure ==
Let <math>\mathfrak g</math> be a complex [[semisimple Lie algebra]].  Let further <math>\mathfrak h</math> be a [[Semisimple_Lie_algebra#Cartan_subalgebras_and_root_systems|Cartan subalgebra]] of <math>\mathfrak g</math>.  Then <math>\mathfrak h</math> acts on <math>\mathfrak g</math> via simultaneously [[diagonalizable]] linear maps in the [[adjoint representation of a Lie algebra|adjoint representation]].  For {{math|''λ''}} in <math>\mathfrak h^*,</math> define the subspace <math>\mathfrak g_\lambda\subset\mathfrak g</math> by

:<math>\mathfrak{g}_\lambda := \{X\in\mathfrak{g}: [H,X]=\lambda(H)X\text{ for all }H\in\mathfrak{h}\}. </math>

We say that <math>\lambda\in\mathfrak h^*</math> is a '''root''' if <math>\lambda\neq 0</math> and the subspace <math>\mathfrak g_\lambda</math> is nonzero. In this case <math>\mathfrak g_\lambda</math> is called the '''root space''' of {{math|''λ''}}. For each root <math>\lambda</math>, the root space <math>\mathfrak g_\lambda</math> is one-dimensional.<ref>{{harvnb|Hall|2015}} Theorem 7.23</ref> Meanwhile, the definition of Cartan subalgebra guarantees that <math>\mathfrak g_0=\mathfrak h</math>.

Let {{math|''R''}} be the set of all roots. Since the elements of <math>\mathfrak h</math> are simultaneously diagonalizable, we have

:<math>\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\lambda\in R}\mathfrak{g}_\lambda.</math>

The [[Cartan subalgebra]] <math>\mathfrak h</math> inherits a nondegenerate bilinear form from the [[Killing form]] on <math>\mathfrak g</math>. This form induces a form on <math>\mathfrak h^*</math> and the restriction of that form to the real span of the roots is an inner product. One can show that with respect to this inner product {{math|''R''}} is a reduced crystallographic root system.<ref>{{harvnb|Hall|2015}} Theorem 7.30</ref>

Let moreover {{math|Δ}} be a choice of [[root system#Positive roots and simple roots|simple roots]]. Now, it is not particularly difficult to find the following set of generators <math>H_\lambda,X_\lambda,Y_\lambda\text{ for }\lambda\in\Delta</math> satisfying the Chevalley–Serre relations<ref>{{harvnb|Humphreys|1973}} Section 18.1</ref>
:<math>\begin{align}[][H_\lambda,H_\mu] &=0 \text{ for all }\lambda,\mu\in\Delta,\\
\left[H_\lambda,X_\mu\right] &= C_{\mu,\lambda}X_\mu,\\
\left[H_\lambda,Y_\mu\right] &= -C_{\mu,\lambda}Y_\mu,\\
\left[X_\mu,Y_\lambda\right] &= \delta_{\mu\lambda}H_\mu,\\
\mathrm{ad}_{X_\lambda}^{1-C_{\mu,\lambda}}(X_\mu) &= 0\text{ for }\lambda\ne\mu,\\
\mathrm{ad}_{Y_\lambda}^{1-C_{\mu,\lambda}}(Y_\mu) &= 0\text{ for }\lambda\ne\mu.\end{align}</math>

Here <math>C_{\lambda,\mu}</math> is the coefficient of the [[Cartan matrix]], given by
:<math>C_{\lambda,\mu}=2\frac{(\lambda,\mu)}{(\mu,\mu)}</math>.
Note that if <math>\lambda</math> and <math>\mu</math> are in <math>\Delta</math> with <math>\lambda\neq\mu</math>, then <math>(\lambda,\mu)\leq 0</math>, so that <math>C_{\lambda,\mu}</math> is a non-positive integer and <math>1-C_{\lambda,\mu}</math> is a positive integer.

.<ref>{{harvnb|Humphreys|1973}} Proposition 18.1</ref> For each simple root <math>\lambda\in\Delta</math>, we can find <math>X_\lambda</math> in the root space <math>\mathfrak g_\lambda</math>, <math>Y_\lambda</math> in the root space <math>\mathfrak g_{-\lambda}</math> and <math>H_\lambda</math> in the Cartan subalgebra satisfying the standard <math>\mathrm{sl}(2;\mathbb C)</math> relations: <math>[H_\lambda,X_\lambda]=2X_\lambda</math>, <math>[H_\lambda,Y_\lambda]=-2Y_\lambda</math>, and <math>[X_\lambda,Y_\lambda]=H_\lambda</math>. These will be our generators.

Now, the element <math>H_\lambda</math> is the ''coroot'' associated to <math>\lambda</math>, which means that after we identify <math>\mathfrak h</math> with its dual, we have <math>H_\lambda=2\lambda/(\lambda,\lambda).</math><ref>{{harvnb|Hall|2015}} Equation (7.9)</ref> Then we have, for example,
:<math>[H_\lambda,X_\mu]=(\mu,H_\lambda)X_\mu=2\frac{(\mu,\lambda)}{(\lambda,\lambda)}X_\mu=C_{\mu,\lambda}X_\mu.</math>
This sort of reasoning verifies the first four relations above.

The last two relations hold because <math>[X_\lambda,X_\mu]</math> belongs to the root space <math>\mathfrak g_{\lambda+\mu}</math>, and more generally, <math>\mathrm{ad}_{X_\lambda}^{k}(X_\mu)</math> belongs to <math>\mathfrak g_{k\lambda+\mu}</math>. But, as we shall see momentarily, if <math>\lambda</math> and <math>
\mu</math> are simple roots, then <math>k\lambda+\mu</math> is not a root if <math>k=1-C_{\mu,\lambda}</math>, so that <math>\mathrm{ad}_{X_\lambda}^{k}(X_\mu)</math> must be zero. To see that <math>k\lambda+\mu</math> is not a root, note that if <math>\lambda</math> and <math>\mu</math> are distinct elements of <math>\Delta</math>, then <math>-\lambda+\mu</math> cannot be a root, for this would violate one of the defining properties of a base—that the expansion of a root in terms of the base cannot have both positive and negative coefficients. But then if <math>s_\lambda</math> is the reflection associated to <math>\lambda</math>, we can easily calculate that 
:<math>s_\lambda\cdot(-\lambda+\mu)=\lambda+\mu-C_{\mu,\lambda}\lambda=k\lambda+\mu</math>.
Then since <math>-\lambda+\mu</math> is not a root, neither is <math>k\lambda+\mu</math>.

Note that the elements <math>\{X_\lambda,Y_\lambda,H_\lambda\},\,\lambda\in\Delta,</math> ''do not'' span <math>\mathfrak g</math> as a vector space, because <math>\lambda</math> does not range over all the positive roots, but only over the base. Nevertheless, these elements generate <math>\mathfrak g</math> as a Lie algebra.<ref>{{harvnb|Humphreys|1973}} Section 18.3</ref>

===Serre's theorem===
{{main|Serre's theorem on a semisimple Lie algebra}}
Serre's theorem asserts the above generators and relations completely determine <math>\mathfrak g</math>. In fact Serre's theorem asserts that starting from an arbitrary root system—not assumed to come from a semisimple Lie algebra—we can use the above relations to ''define'' a Lie algebra, the Lie algebra is finite-dimensional and semisimple, and the root system of that Lie algebra is the root system <math>R</math> we started from.<ref>{{harvnb|Humphreys|1973}} Section 18.3</ref>

A consequence of Serre's theorem is this: 
*Every (reduced, crystallographic) root system comes from a semisimple Lie algebra.-->