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{{Lie groups |Semi-simple}}
In [[Mathematics|mathematics]], there is a one-to-one correspondence between reduced crystallographic [[Root system|root system]]s and [[Semisimple Lie algebra|semisimple Lie algebra]]s.  Here the construction of a '''root system of a semisimple Lie algebra'''{{spaced ndash}}and, conversely, the construction of a semisimple Lie algebra from a reduced crystallographic root system{{spaced ndash}}are shown.

==Associating a root system to a semisimple Lie algebra==

Let <math>\mathfrak g</math> be a complex [[Semisimple Lie algebra|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.  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>

==Serre's relations: Associating a semisimple Lie algebra to a root system==
===The relations===
Let {{math|''E''}} be a [[Euclidean space]] and {{math|''R''}} a reduced crystallographic root system in {{math|''E''}}.  Let moreover {{math|Δ}} be a choice of [[Root system#Positive roots and simple roots|simple roots]].  We define a complex Lie algebra over the generators

:<math>H_\lambda,X_\lambda,Y_\lambda\text{ for }\lambda\in\Delta</math>

with 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.

===Finding generators with these relations===
Now, if we are ''given'' a semisimple Lie algebra <math>\mathfrak g</math> with root system <math>R</math>, it is not particularly difficult to find a set of generators for <math>\mathfrak g</math> satisfying the above relations.<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===
Serre's theorem asserts much more than just the existence of such generators for a given semisimple Lie algebra <math>\mathfrak g</math>. First, the claim is that the above relations completely determine <math>\mathfrak g</math>; that is, there are no other relations in <math>\mathfrak g</math> besides ones that follow from these. Even more, however, 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.

== Application to the classification of semisimple Lie algebras ==
The preceding results help in the process of reducing the classification of semisimple Lie algebras to the task of classifying reduced crystallographic root systems, which is then done in terms of [[Dynkin diagram]]s. Although the above describes how to construct a root system from a Lie algebra and vice versa, there are still two things to prove before we can obtain a one-to-one correspondence. 
*First, we need to know that each Lie algebra gives only one root system (up to isomorphism). This is shown by proving that the Cartan subalgebra of a semisimple Lie algebra <math>\mathfrak g</math> is unique up to automorphism.<ref>{{harvnb|Humphreys|1973}} Corollary 16.4</ref> It follows that all Cartan subalgebras of <math>\mathfrak g</math> give isomorphic root systems.
*Second, we must prove that each root system gives only one Lie algebra (up to isomorphism). That is, we must show that if two Lie algebras have isomorphic root systems, the Lie algebras are also isomorphic.<ref>{{harvnb|Humphreys|1973}} Theorem 14.2</ref>

==See also==
*[[Semisimple Lie algebra]]
*[[Root system]]

== Notes ==
{{reflist}}

== References ==
* {{Citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
*{{Citation|last=Humphreys|first=James|title=Introduction to Lie Algebras and Representation Theory|series=Graduate Texts in Mathematics|volume=9|publisher=Springer|year=1973}}
* V.S. Varadarajan, ''Lie groups, Lie algebras, and their representations'', GTM, Springer 1984.

==External links==
* {{springer|title=Coxeter group|id=p/c026980}}
* {{MathWorld | urlname=CoxeterGroup  | title=Coxeter group }}
* {{Citation|url=http://www.jenn3d.org/index.html|title= Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators}}
* {{Citation | chapter = Weyl group | title = Encyclopaedia of Mathematics | publisher = SpringerLink | first1 = V.L. | last1 = Popov |  first2 = A.S. | last2 = Fedenko | year = 2001 | url = https://www.encyclopediaofmath.org/index.php/Lie_algebra,_semi-simple }}

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