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Difference between revisions of "Lie algebra, semi-simple"

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A Lie algebra that has no non-zero solvable ideals (see [[Lie algebra, solvable|Lie algebra, solvable]]). Henceforth finite-dimensional semi-simple Lie algebras over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585101.png" /> of characteristic 0 are considered (for semi-simple Lie algebras over a field of non-zero characteristic see [[Lie algebra|Lie algebra]]).
+
A Lie algebra that has no non-zero solvable ideals (see [[Lie algebra, solvable|Lie algebra, solvable]]). Henceforth finite-dimensional semi-simple Lie algebras over a field $  k $  of characteristic 0 are considered (for semi-simple Lie algebras over a field of non-zero characteristic see [[Lie algebra|Lie algebra]]).
  
The fact that a finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585102.png" /> is semi-simple is equivalent to any of the following conditions:
+
The fact that a finite-dimensional Lie algebra $  \mathfrak g $  is semi-simple is equivalent to any of the following conditions:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585103.png" /> does not contain non-zero Abelian ideals;
+
1) $  \mathfrak g $  does not contain non-zero Abelian ideals;
  
2) the [[Killing form|Killing form]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585104.png" /> is non-singular (Cartan's criterion);
+
2) the [[Killing form|Killing form]] of $  \mathfrak g $  is non-singular (Cartan's criterion);
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585105.png" /> splits into the direct sum of non-Abelian simple ideals;
+
3) $  \mathfrak g $  splits into the direct sum of non-Abelian simple ideals;
  
4) every finite-dimensional linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585106.png" /> is completely reducible (in other words: every finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585107.png" />-module is semi-simple);
+
4) every finite-dimensional linear representation of $  \mathfrak g $  is completely reducible (in other words: every finite-dimensional $  \mathfrak g $ -module is semi-simple);
  
5) the one-dimensional cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585108.png" /> with values in an arbitrary finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l0585109.png" />-module is trivial.
+
5) the one-dimensional cohomology of $  \mathfrak g $  with values in an arbitrary finite-dimensional $  \mathfrak g $ -module is trivial.
  
 
Any ideal and any quotient algebra of a semi-simple Lie algebra is also semi-simple. The decomposition of a semi-simple Lie algebra mentioned in condition 3) is unique. A special case of condition 5) is the following assertion: All derivations of a semi-simple Lie algebra are inner. The property of a Lie algebra of being semi-simple is preserved by both extensions and restrictions of the ground field.
 
Any ideal and any quotient algebra of a semi-simple Lie algebra is also semi-simple. The decomposition of a semi-simple Lie algebra mentioned in condition 3) is unique. A special case of condition 5) is the following assertion: All derivations of a semi-simple Lie algebra are inner. The property of a Lie algebra of being semi-simple is preserved by both extensions and restrictions of the ground field.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851010.png" /> be a semi-simple Lie algebra over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851011.png" />. The adjoint representation maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851012.png" /> isomorphically onto the linear Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851013.png" />, which is the Lie algebra of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851014.png" /> of all automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851015.png" /> and is therefore an algebraic Lie algebra (cf. [[Lie algebra, algebraic|Lie algebra, algebraic]]). An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851016.png" /> is said to be semi-simple (nilpotent) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851017.png" /> is semi-simple (respectively, nilpotent). This property of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851018.png" /> is preserved by any homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851019.png" /> into another semi-simple Lie algebra. The identity component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851020.png" /> coincides with the group of inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851021.png" />, that is, it is generated by the automorphisms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851023.png" />.
+
Let $  \mathfrak g $  be a semi-simple Lie algebra over an algebraically closed field $  k $ . The adjoint representation maps $  \mathfrak g $  isomorphically onto the linear Lie algebra $  \mathop{\rm ad}\nolimits \  \mathfrak g $ , which is the Lie algebra of the algebraic group $  \mathop{\rm Aut}\nolimits \  \mathfrak g $  of all automorphisms of $  \mathfrak g $  and is therefore an algebraic Lie algebra (cf. [[Lie algebra, algebraic|Lie algebra, algebraic]]). An element $  X \in \mathfrak g $  is said to be semi-simple (nilpotent) if $  \mathop{\rm ad}\nolimits \  X $  is semi-simple (respectively, nilpotent). This property of an element $  X $  is preserved by any homomorphism of $  \mathfrak g $  into another semi-simple Lie algebra. The identity component $  (  \mathop{\rm Aut}\nolimits \  \mathfrak g ) ^{0} $  coincides with the group of inner automorphisms of $  \mathfrak g $ , that is, it is generated by the automorphisms of the form $  \mathop{\rm exp}\nolimits (  \mathop{\rm ad}\nolimits \  X ) $ , $  X \in \mathfrak g $ .
  
In the study of semi-simple Lie algebras over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851024.png" /> an important role is played by the roots of a semi-simple Lie algebra, which are defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851025.png" /> be a [[Cartan subalgebra|Cartan subalgebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851026.png" />. For a non-zero linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851027.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851028.png" /> denote the linear subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851029.png" /> given by the condition
+
In the study of semi-simple Lie algebras over an algebraically closed field $  k $  an important role is played by the roots of a semi-simple Lie algebra, which are defined as follows. Let $  \mathfrak h $  be a [[Cartan subalgebra|Cartan subalgebra]] of $  \mathfrak g $ . For a non-zero linear function $  \alpha \in \mathfrak h ^ \ast  $ , let $  \mathfrak g _ \alpha  $  denote the linear subspace of $  \mathfrak g $  given by the condition $$
 +
\mathfrak g _ \alpha    =
 +
\{ {X \in \mathfrak g} : {[ H ,\  X ] = \alpha (H) X ,  H \in \mathfrak h} \}
 +
.
 +
$$ If  $  \mathfrak g _ \alpha  \neq 0 $ , then  $  \alpha $  is called a root of  $  \mathfrak g $  with respect to  $  \mathfrak h $ . The set  $  \Sigma $  of all non-zero roots is called the root system, or system of roots, of  $  \mathfrak g $ . One has the root decomposition $$
 +
\mathfrak g  =  \mathfrak h +
 +
\sum _ {\alpha \in \Sigma}
 +
\mathfrak g _ \alpha  .
 +
$$ The root system and the root decomposition of a semi-simple Lie algebra have the following properties:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851030.png" /></td> </tr></table>
+
a)  $  \Sigma $  generates  $  \mathfrak h ^ \ast  $  and is a reduced [[Root system|root system]] in the abstract sense (in the linear hull of  $  \Sigma $  over the field of the real numbers). The system  $  \Sigma $  is irreducible if and only if  $  \mathfrak g $  is simple.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851032.png" /> is called a root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851033.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851034.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851035.png" /> of all non-zero roots is called the root system, or system of roots, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851036.png" />. One has the root decomposition
+
b) For any  $  \alpha \in \Sigma $ , $$
 +
\mathop{\rm dim}\nolimits \  \mathfrak g _ \alpha    =
 +
\mathop{\rm dim}\nolimits \  [ \mathfrak g _ \alpha  ,\  \mathfrak g _ {- \alpha} ]  =   1 .
 +
$$ There is a unique element  $  H _ \alpha  \in [ \mathfrak g _ \alpha  ,\  \mathfrak g _ {- \alpha} ] $  such that  $  \alpha ( H _ \alpha  ) = 2 $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851037.png" /></td> </tr></table>
+
c) For every non-zero  $  X _ \alpha  \in \mathfrak g _ \alpha  $  there is a unique  $  Y _ \alpha  \in \mathfrak g _ {- \alpha} $  such that  $  [ X _ \alpha  ,\  Y _ \alpha  ] = H _ \alpha  $ , and $$
 +
[ H _ \alpha  ,\  X _ \alpha  ]  =   2 X _ \alpha   
 +
\textrm{ and }  [ H _ \alpha  ,\  Y _ \alpha  ]  =   -
 +
2 Y _ \alpha  .
 +
$$ Moreover, $$
 +
\beta ( H _ \alpha  )  =
  
The root system and the root decomposition of a semi-simple Lie algebra have the following properties:
+
\frac{2 ( \alpha ,\  \beta )}{( \alpha ,\  \alpha )}
 +
 +
\alpha ,\  \beta \in \Sigma ,
 +
$$ where  $  ( \  ,\  ) $  is the scalar product induced by the Killing form.
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851038.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851039.png" /> and is a reduced [[Root system|root system]] in the abstract sense (in the linear hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851040.png" /> over the field of the real numbers). The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851041.png" /> is irreducible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851042.png" /> is simple.
+
d) If  $  \alpha ,\  \beta \in \Sigma $  and  $  \alpha + \beta \neq 0 $ , then  $  \mathfrak g _ \alpha  $  and  $  \mathfrak g _ \beta  $  are orthogonal with respect to the Killing form and [ \mathfrak g _ \alpha  ,\  \mathfrak g _ \beta  ] = \mathfrak g _ {\alpha + \beta} $ .
  
b) For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851043.png" />,
+
A basis  $  \{ \alpha _{1} \dots \alpha _{n} \} $  of the root system  $  \Sigma $  is also called a system of simple roots of the algebra  $  \mathfrak g $ . Let  $  \Sigma _{+} $  be the system of positive roots with respect to the given basis and let  $  X _ {- \alpha} = Y _ \alpha  $  $  ( \alpha \in \Sigma _{+} ) $ . Then the elements $$
 +
H _ {\alpha _{1}} \dots H _ {\alpha _{k}} ,\
 +
X _ \alpha    ( \alpha \in \Sigma )
 +
$$ form a basis of  $  \mathfrak g $ , called a Cartan basis. On the other hand, the elements $$
 +
X _ {\alpha _{i}} ,
 +
X _ {- \alpha _{i}} 
 +
( i = 1 \dots n )
 +
$$ form a system of generators of  $  \mathfrak g $ , and the defining relations have the following form: $$
 +
[ [ X _ {\alpha _{i}} ,\  X _ {- \alpha _{i}} ] ,\
 +
X _ {\alpha _{j}} ]  =  n ( i ,\  j )
 +
X _ {\alpha _{j}} ,
 +
$$  $$
 +
[ [ X _ {\alpha _{i}} ,\  X _ {- \alpha _{i}} ]
 +
,\  X _ {- \alpha _{j}} ]  =   - n ( i ,\  j ) X _ {\alpha _{j}} ,
 +
$$  $$
 +
(  \mathop{\rm ad}\nolimits \  X _ {\alpha _{i}} ) ^ {1 - n
 +
( i ,\  j )} X _ {\alpha _{j}}  =  0 ,
 +
$$  $$
 +
(  \mathop{\rm ad}\nolimits \  X _ {- \alpha _{i}} ) ^ {1 - n
 +
( i ,\  j )} X _ {- \alpha _{j}}  =   0 .
 +
$$ Here  $  i ,\  j = 1 \dots n $  and $$
 +
n ( i ,\  j )  =  \alpha _{j}
 +
( H _{i} )  = 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851044.png" /></td> </tr></table>
+
\frac{2 ( \alpha _{i} ,\  \alpha _{j} )}{( \alpha _{j} ,\  \alpha _{j} )}
 +
.
 +
$$ Property d) implies that $$
 +
[ X _ \alpha  ,\  X _ \beta  ]  =
 +
\left \{
  
There is a unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851046.png" />.
+
\begin{array}{ll}
 +
N _ {\alpha , \beta} X _ {\alpha + \beta}  &  \textrm{ if }  \alpha + \beta \in \Sigma ,  \\
 +
0 &  \textrm{ if }  \alpha + \beta \notin \Sigma ,  \\
 +
\end{array}
  
c) For every non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851047.png" /> there is a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851049.png" />, and
+
\right .$$where  $  N _ {\alpha , \beta} \in k $ . The elements  $  X _ \alpha  $  $  ( \alpha \in \Sigma _{+} ) $  can be chosen in such a way that $$
 +
N _ {\alpha , \beta}  =   - N _ {- \alpha , - \beta} 
 +
\textrm{ and }  N _ {\alpha , \beta}  =   \pm ( p + 1 ) ,
 +
$$ where  $  p $  is the largest integer such that $  \beta - p \alpha \in \Sigma $ . The corresponding Cartan basis is called a Chevalley basis. The structure constants of  $  \mathfrak g $  in this basis are integers, which makes it possible to associate with  $  \mathfrak g $  Lie algebras and algebraic groups (see [[Chevalley group|Chevalley group]]) over fields of arbitrary characteristic. If  $  k = \mathbf C $ , then the linear hull over  $  \mathbf R $  of the vectors $$
 +
i H _ \alpha  ,  X _ \alpha  - X _ {- \alpha} , 
 +
i ( X _ \alpha  + X _ {- \alpha} ) 
 +
\quad ( \alpha \in \Sigma _{+} )
 +
$$ is a compact real form of  $  \mathfrak g $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851050.png" /></td> </tr></table>
+
A semi-simple Lie algebra is defined up to an isomorphism by its Cartan subalgebra and the corresponding root system. More precisely, if  $  \mathfrak g _{1} $  and  $  \mathfrak g _{2} $  are semi-simple Lie algebras over  $  k $ ,  $  \mathfrak h _{1} $  and  $  \mathfrak h _{2} $  are their Cartan subalgebras and  $  \Sigma _{1} $  and  $  \Sigma _{2} $  are the corresponding root systems, then every isomorphism  $  \mathfrak h _{1} \rightarrow \mathfrak h _{2} $  that induces an isomorphism of the root systems  $  \Sigma _{1} $  and  $  \Sigma _{2} $  can be extended to an isomorphism  $  \mathfrak g _{1} \rightarrow \mathfrak g _{2} $ . On the other hand, any reduced root system can be realized as the root system of some semi-simple Lie algebra. Thus, the classification of semi-simple Lie algebras (respectively, simple non-Abelian Lie algebras) over an algebraically closed field  $  k $  essentially coincides with the classification of reduced root systems (respectively, irreducible reduced root systems).
  
Moreover,
+
Simple Lie algebras that correspond to root systems of types  $  A $ – $  D $  are said to be classical and have the following form.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851051.png" /></td> </tr></table>
+
Type  $  A _{n} $ ,  $  n \geq 1 $ .  $  \mathfrak g = \mathfrak s \mathfrak l ( n + 1 ,\  k ) $ , the algebra of linear transformations of the space  $  k ^{n+1} $  with trace 0; $  \mathop{\rm dim}\nolimits \  \mathfrak g = n ( n + 2 ) $ .
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851052.png" /> is the scalar product induced by the Killing form.
+
Type  $  B _{n} $ ,  $  n \geq 2 $ .  $  \mathfrak g = \mathfrak s \mathfrak o ( 2 n + 1 ,\  k ) $ , the algebra of linear transformations of the space  $  k ^{2n+1} $  that are skew-symmetric with respect to a given non-singular symmetric bilinear form;  $  \mathop{\rm dim}\nolimits \  \mathfrak g = n ( 2 n + 1 ) $ .
  
d) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851056.png" /> are orthogonal with respect to the Killing form and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851057.png" />.
+
Type  $  C _{n} $ ,  $  n \geq 3 $ . $  \mathfrak g = \mathfrak s \mathfrak p ( n ,\  k ) $ , the algebra of linear transformations of the space  $  k ^{2n} $  that are skew-symmetric with respect to a given non-singular skew-symmetric bilinear form;  $  \mathop{\rm dim}\nolimits \  \mathfrak g = n ( 2 n + 1 ) $ .
  
A basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851058.png" /> of the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851059.png" /> is also called a system of simple roots of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851060.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851061.png" /> be the system of positive roots with respect to the given basis and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851062.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851063.png" />. Then the elements
+
Type  $  D _{n} $ ,  $  n \geq 4 $ .  $  \mathfrak g = \mathfrak s \mathfrak o ( 2 n ,\  k ) $ , the algebra of linear transformations of the space  $  k ^{2n} $  that are skew-symmetric with respect to a given non-singular symmetric bilinear form;  $  \mathop{\rm dim}\nolimits \  \mathfrak g = n ( 2 n - 1 ) $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851064.png" /></td> </tr></table>
+
The simple Lie algebras corresponding to the root systems of types  $  E _{6} $ ,  $  E _{7} $ ,  $  E _{8} $ ,  $  F _{4} $ ,  $  G _{2} $  are called special, or exceptional (see [[Lie algebra, exceptional|Lie algebra, exceptional]]).
  
form a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851065.png" />, called a Cartan basis. On the other hand, the elements
+
The [[Cartan matrix|Cartan matrix]] of a semi-simple Lie algebra over an algebraically closed field also determines this algebra uniquely up to an isomorphism. The Cartan matrices of the simple Lie algebras have the following form: $$
 +
A _{n} : 
 +
\left \|
 +
\begin{array}{rrrcr}
 +
2  &-1  & 0 &\dots  & 0  \\
 +
-1  & 2  &-1  &\dots  & 0  \\
 +
0  &-1  & 2  &\dots  & 0  \\
 +
.  & .  & .  &\ddots  & .  \\
 +
0  & 0  & 0  &\dots  &-1  \\
 +
0  & 0  & 0  &\dots  & 2  \\
 +
\end{array}
 +
\right \| ,
 +
$$  $$
 +
B _{n} :   \left \|
 +
\begin{array}{rrrrrr}
 +
2  &-1  & 0  &\dots  & 0  & 0  \\
 +
-1  & 2  &-1  &\dots  & 0  & 0  \\
 +
0  &-1  & 2  &\dots  & 0  & 0  \\
 +
.  & .  & . &\ddots & . & . \\
 +
0  & 0  & 0  &\dots  & 2  &-2  \\
 +
0  & 0  & 0  &\dots  &-1  & 2  \\
 +
\end{array}
 +
\right \| ,
 +
$$  $$
 +
C _{n} :    \left \|
 +
\begin{array}{rrrrrrr}
 +
2  &-1  & 0  &\dots  & 0  & 0  \\
 +
-1  & 2  &-1  &\dots  & 0  & 0  \\
 +
0  &-1  & 2  &\dots  & 0  & 0  \\
 +
.  & .  & .  &\cdots & .  & .  \\
 +
0  & 0  & 0  &\dots  & 2  &-1  \\
 +
0  & 0  & 0  &\dots  &-2  & 2  \\
 +
\end{array}
 +
\right \| ,
 +
$$  $$
 +
D _{n} :    \left \|
 +
\begin{array}{rrrrrrrr}
 +
2  &-1  & 0  &\dots  & 0  & 0  & 0  & 0  \\
 +
-1  & 2  &-1  &\dots  & 0  & 0  & 0  & 0  \\
 +
0  &-1  & 2  &\dots  & 0  & 0  & 0  & 0  \\
 +
.  & .  & .  &\ddots & .  & .  & .  & . \\
 +
0  & 0  & 0  &\dots  & 2  &-1  & 0  & 0  \\
 +
0  & 0  & 0  &\dots  &-1  & 2  &-1  &-1  \\
 +
0  & 0  & 0  &\dots  & 0  &-1  & 2  & 0  \\
 +
0  & 0  & 0  &\dots  & 0  &-1  & 0  & 2  \\
 +
\end{array}
 +
\right \| ,
 +
$$  $$
 +
E _{6} :    \left \|
 +
\begin{array}{rrrrrr}
 +
2  & 0  &-1  & 0  & 0  & 0  \\
 +
0  & 2  & 0  &-1  & 0  & 0  \\
 +
-1  & 0  & 2  &-1  & 0  & 0  \\
 +
0  &-1  &-1  & 2  &-1  & 0  \\
 +
0  & 0  & 0  &-1  & 2  &-1  \\
 +
0  & 0  & 0  & 0  &-1  & 2  \\
 +
\end{array}
 +
\right \| ,
 +
$$  $$
 +
E _{7} :    \left \|
 +
\begin{array}{rrrrrrr}
 +
2  & 0  &-1  & 0  & 0  & 0  & 0  \\
 +
0  & 2  & 0  &-1  & 0  & 0  & 0  \\
 +
-1  & 0  & 2  &-1  & 0  & 0  & 0  \\
 +
0  &-1  &-1  & 2  &-1  & 0  & 0  \\
 +
0  & 0  & 0  &-1  & 2  &-1  & 0  \\
 +
0  & 0  & 0  & 0  &-1  & 2  &-1  \\
 +
0  & 0  & 0  & 0  & 0  &-1  & 2  \\
 +
\end{array}
 +
\right \| ,
 +
$$  $$
 +
E _{8} :    \left \|
 +
\begin{array}{rrrrrrrr}
 +
2  & 0  &-1  & 0  & 0  & 0  & 0  & 0  \\
 +
0  & 2  & 0  &-1  & 0  & 0  & 0  & 0  \\
 +
-1  & 0  & 2  &-1  & 0  & 0  & 0  & 0  \\
 +
0  &-1  &-1  & 2  &-1  & 0  & 0  & 0  \\
 +
0  & 0  & 0  &-1  & 2  &-1  & 0  & 0  \\
 +
0  & 0  & 0  & 0  &-1  & 2  &-1  & 0  \\
 +
0  & 0  & 0  & 0  & 0  &-1  & 2  &-1  \\
 +
0  & 0  & 0  & 0  & 0  & 0  &-1  & 2  \\
 +
\end{array}
 +
\right \| ,
 +
$$  $$
 +
F _{4} :    \left \|
 +
\begin{array}{rrrr}
 +
2  &-1  & 0  & 0  \\
 +
-1  & 2  &-2  & 0  \\
 +
0  &-1  & 2  &-1  \\
 +
0  & 0  &-1  & 2  \\
 +
\end{array}
 +
\right \| ,     G _{2} :    \left \|
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851066.png" /></td> </tr></table>
+
\begin{array}{rr}
 +
2  &-1  \\
 +
-3  & 2  \\
 +
\end{array}
 +
\right \| .
 +
$$ The classification of split semi-simple Lie algebras over an arbitrary field  $  k $  of characteristic zero (a semi-simple Lie algebra  $  \mathfrak g $  is said to be split if it has a Cartan subalgebra  $  \mathfrak h \subset \mathfrak g $  such that all characteristic roots of the operators  $  \mathop{\rm ad}\nolimits \  X $ ,  $  X \in \mathfrak h $ , lie in  $  k $ ) goes in the same way as in the case of an algebraically closed field. Namely, to every irreducible reduced root system corresponds a unique split semi-simple Lie algebra. In particular, split semi-simple Lie algebras of types  $  A $ – $  D $  have the form stated above, except that in the cases  $  B $  and  $  D $  one must consider non-singular symmetric bilinear forms with Witt index  $  n $ .
  
form a system of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851067.png" />, and the defining relations have the following form:
+
The problem of classifying arbitrary semi-simple Lie algebras over  $  k $  reduces to the following problem: To list, up to an isomorphism, all  $  k $ -forms  $  \mathfrak g _{0} \subset \mathfrak g $ , that is, all  $  k $ -subalgebras  $  \mathfrak g _{0} \subset \mathfrak g $  such that  $  \mathfrak g = \mathfrak g _{0} \otimes _{k} K $ . Here  $  K $  is an algebraically closed extension of  $  k $  and  $  \mathfrak g $  is a given semi-simple Lie algebra over  $  K $ . The solution of this problem can also be obtained in terms of root systems (see [[Form of an algebraic group|Form of an algebraic group]]; [[Form of an (algebraic) structure|Form of an (algebraic) structure]]). When  $  \mathfrak g $  is a classical simple Lie algebra over  $  k $  (other than  $  D _{4} $ ), there is another method of classifying  $  k $ -forms in  $  \mathfrak g $ , based on an examination of simple associative algebras (see [[#References|[3]]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851068.png" /></td> </tr></table>
+
When  $  k = \mathbf R $  the classification of semi-simple Lie algebras goes as follows (see [[#References|[6]]], [[#References|[7]]]). Every simple non-Abelian Lie algebra over  $  \mathbf R $  is either a simple Lie algebra over  $  \mathbf C $  (regarded as an algebra over  $  \mathbf R $ ), or the real form of a simple Lie algebra over  $  \mathbf C $ . The classification of real forms  $  \mathfrak g _{0} $  in a simple classical Lie algebra  $  \mathfrak g $  over  $  \mathbf C $  is as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851069.png" /></td> </tr></table>
+
I) Type  $  A _{n} $ :  $  \mathfrak g = \mathfrak s \mathfrak l ( n + 1 ,\  \mathbf C ) $ ,  $  n \geq 1 $ .  $  A _{I} $ :  $  \mathfrak g _{0} = \mathfrak s \mathfrak l ( n + 1 ,\  \mathbf R ) $ .  $  A _{II} $ : $  n + 1 = 2 m $  is even,  $  \mathfrak g _{0} = \mathfrak s \mathfrak u ^ \ast  ( 2 n ) $ , the subalgebra of elements of  $  \mathfrak s \mathfrak l ( 2 m ,\  \mathbf C ) $  that preserve a certain quaternion structure.  $  A _{III} $ : $  \mathfrak g _{0} = \mathfrak s \mathfrak u ( p ,\  n + 1 - p ) $ , the subalgebra of elements of  $  \mathfrak s \mathfrak l ( n + 1 ,\  \mathbf C ) $  that are skew-symmetric with respect to a non-singular Hermitian form of positive index  $  p $ ,  $  0 \leq p \leq ( n + 1 ) / 2 . $
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851070.png" /></td> </tr></table>
+
II) Type  $  B _{n} $ :  $  \mathfrak g = \mathfrak s \mathfrak o ( 2 n + 1 ,\  \mathbf C ) $ ,  $  n \geq 2 $ .  $  B _{I} $ : $  \mathfrak g _{0} = \mathfrak s \mathfrak o ( p ,\  2 n + 1 - p ) $ , the algebra of a linear transformations of the space  $  \mathbf R ^{2n+1} $  that are skew-symmetric with respect to a non-singular symmetric bilinear form of positive index  $  p $ ,  $  0 \leq p \leq n $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851071.png" /></td> </tr></table>
+
III) Type  $  C _{n} $ :  $  \mathfrak g = \mathfrak s \mathfrak p (n,\  \mathbf C ) $ ,  $  n \geq 3 $ .  $  C _{I} $ : $  \mathfrak g _{0} = \mathfrak s \mathfrak p ( n ,\  \mathbf R ) $ , the algebra of linear transformations of the space  $  \mathbf R ^{2n} $  that are skew-symmetric with respect to a non-singular skew-symmetric bilinear form.  $  C _{II} $ : $  \mathfrak g _{0} = \mathfrak s \mathfrak p ( p ,\  n - p ) $ ,  $  0 \leq p \leq n / 2 $ , the subalgebra of  $  \mathfrak s \mathfrak u ( 2 p ,\  2 ( n - p ) ) $  consisting of transformations that preserve a certain quaternion structure.
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851072.png" /> and
+
IV) Type  $  D _{n} $ :  $  \mathfrak g = \mathfrak s \mathfrak o ( 2 n ,\  \mathbf C ) $ ,  $  n \geq 4 $ .  $  D _{I} $ :  $  \mathfrak g _{0} = \mathfrak s \mathfrak o ( p ,\  2 n - p ) $ , the algebra of linear transformations of the space  $  \mathbf R ^{2n} $  that are skew-symmetric with respect to a non-singular bilinear symmetric form of positive index  $  p $ ,  $  0 \leq p \leq n $ .  $  D _{III} $ :  $  \mathfrak g _{0} = \mathfrak s \mathfrak o ^ \ast  ( 2 n ,\  \mathbf C ) $ , the subalgebra of  $  \mathfrak s \mathfrak o (2n ,\  \mathbf C ) $  consisting of transformations that preserve a certain quaternion structure.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851073.png" /></td> </tr></table>
+
Semi-simple Lie algebras over the field $  \mathbf C $  were first considered in papers by W. Killing , who gave a classification of them, although in his proofs there were gaps, which were filled by E. Cartan [[#References|[2]]]. In the papers of Killing and Cartan the roots of a Lie algebra appeared as the characteristic roots of the operator $  \mathop{\rm ad}\nolimits \  X $ . Cartan also gave a classification of real semi-simple Lie algebras by establishing a deep connection between these algebras and globally symmetric Riemannian spaces (cf. [[Globally symmetric Riemannian space|Globally symmetric Riemannian space]]).
 
 
Property d) implies that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851074.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851075.png" />. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851076.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851077.png" /> can be chosen in such a way that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851078.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851079.png" /> is the largest integer such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851080.png" />. The corresponding Cartan basis is called a Chevalley basis. The structure constants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851081.png" /> in this basis are integers, which makes it possible to associate with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851082.png" /> Lie algebras and algebraic groups (see [[Chevalley group|Chevalley group]]) over fields of arbitrary characteristic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851083.png" />, then the linear hull over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851084.png" /> of the vectors
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851085.png" /></td> </tr></table>
 
 
 
is a compact real form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851086.png" />.
 
 
 
A semi-simple Lie algebra is defined up to an isomorphism by its Cartan subalgebra and the corresponding root system. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851088.png" /> are semi-simple Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851091.png" /> are their Cartan subalgebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851093.png" /> are the corresponding root systems, then every isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851094.png" /> that induces an isomorphism of the root systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851096.png" /> can be extended to an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851097.png" />. On the other hand, any reduced root system can be realized as the root system of some semi-simple Lie algebra. Thus, the classification of semi-simple Lie algebras (respectively, simple non-Abelian Lie algebras) over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851098.png" /> essentially coincides with the classification of reduced root systems (respectively, irreducible reduced root systems).
 
 
 
Simple Lie algebras that correspond to root systems of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851099.png" />–<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510100.png" /> are said to be classical and have the following form.
 
 
 
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510102.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510103.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510104.png" /> with trace 0; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510105.png" />.
 
 
 
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510107.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510108.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510109.png" /> that are skew-symmetric with respect to a given non-singular symmetric bilinear form; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510110.png" />.
 
 
 
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510112.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510113.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510114.png" /> that are skew-symmetric with respect to a given non-singular skew-symmetric bilinear form; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510115.png" />.
 
 
 
Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510117.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510118.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510119.png" /> that are skew-symmetric with respect to a given non-singular symmetric bilinear form; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510120.png" />.
 
 
 
The simple Lie algebras corresponding to the root systems of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510125.png" /> are called special, or exceptional (see [[Lie algebra, exceptional|Lie algebra, exceptional]]).
 
 
 
The [[Cartan matrix|Cartan matrix]] of a semi-simple Lie algebra over an algebraically closed field also determines this algebra uniquely up to an isomorphism. The Cartan matrices of the simple Lie algebras have the following form:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510126.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510128.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510130.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510131.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510132.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510133.png" /></td> </tr></table>
 
 
 
The classification of split semi-simple Lie algebras over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510134.png" /> of characteristic zero (a semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510135.png" /> is said to be split if it has a Cartan subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510136.png" /> such that all characteristic roots of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510138.png" />, lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510139.png" />) goes in the same way as in the case of an algebraically closed field. Namely, to every irreducible reduced root system corresponds a unique split semi-simple Lie algebra. In particular, split semi-simple Lie algebras of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510140.png" />–<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510141.png" /> have the form stated above, except that in the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510143.png" /> one must consider non-singular symmetric bilinear forms with Witt index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510144.png" />.
 
 
 
The problem of classifying arbitrary semi-simple Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510145.png" /> reduces to the following problem: To list, up to an isomorphism, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510146.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510147.png" />, that is, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510148.png" />-subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510149.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510150.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510151.png" /> is an algebraically closed extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510153.png" /> is a given semi-simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510154.png" />. The solution of this problem can also be obtained in terms of root systems (see [[Form of an algebraic group|Form of an algebraic group]]; [[Form of an (algebraic) structure|Form of an (algebraic) structure]]). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510155.png" /> is a classical simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510156.png" /> (other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510157.png" />), there is another method of classifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510158.png" />-forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510159.png" />, based on an examination of simple associative algebras (see [[#References|[3]]]).
 
 
 
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510160.png" /> the classification of semi-simple Lie algebras goes as follows (see [[#References|[6]]], [[#References|[7]]]). Every simple non-Abelian Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510161.png" /> is either a simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510162.png" /> (regarded as an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510163.png" />), or the real form of a simple Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510164.png" />. The classification of real forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510165.png" /> in a simple classical Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510166.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510167.png" /> is as follows:
 
 
 
I) Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510168.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510170.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510171.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510172.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510173.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510174.png" /> is even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510175.png" />, the subalgebra of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510176.png" /> that preserve a certain quaternion structure. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510177.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510178.png" />, the subalgebra of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510179.png" /> that are skew-symmetric with respect to a non-singular Hermitian form of positive index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510181.png" />
 
 
 
II) Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510182.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510183.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510184.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510185.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510186.png" />, the algebra of a linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510187.png" /> that are skew-symmetric with respect to a non-singular symmetric bilinear form of positive index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510188.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510189.png" />.
 
 
 
III) Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510190.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510191.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510192.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510193.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510194.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510195.png" /> that are skew-symmetric with respect to a non-singular skew-symmetric bilinear form. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510196.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510197.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510198.png" />, the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510199.png" /> consisting of transformations that preserve a certain quaternion structure.
 
 
 
IV) Type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510200.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510202.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510203.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510204.png" />, the algebra of linear transformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510205.png" /> that are skew-symmetric with respect to a non-singular bilinear symmetric form of positive index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510207.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510208.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510209.png" />, the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510210.png" /> consisting of transformations that preserve a certain quaternion structure.
 
 
 
Semi-simple Lie algebras over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510211.png" /> were first considered in papers by W. Killing , who gave a classification of them, although in his proofs there were gaps, which were filled by E. Cartan [[#References|[2]]]. In the papers of Killing and Cartan the roots of a Lie algebra appeared as the characteristic roots of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510212.png" />. Cartan also gave a classification of real semi-simple Lie algebras by establishing a deep connection between these algebras and globally symmetric Riemannian spaces (cf. [[Globally symmetric Riemannian space|Globally symmetric Riemannian space]]).
 
  
 
====References====
 
====References====
Line 137: Line 226:
  
 
====Comments====
 
====Comments====
The defining relations, mentioned above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510213.png" />, are known as the Serre relations.
+
The defining relations, mentioned above, $  (  \mathop{\rm ad}\nolimits \  X _ {\alpha _{i}} ) ^ {1 - n ( i , j )} ( X _ {\alpha _{j}} ) = 0 $ , are known as the Serre relations.
  
It is customary to encode the information contained in the Cartan matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510214.png" /><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510215.png" /> by means of the so-called Dynkin diagrams.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510216.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
It is customary to encode the information contained in the Cartan matrices $  A _{n} $ $  G _{2} $  by means of the so-called Dynkin diagrams.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _{n} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
</td> <td colname="3" style="background-color:white;" colspan="1">(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510217.png" /> nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510218.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> <td colname="3" style="background-color:white;" colspan="1">( $  n $  nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  B _{n} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
</td> <td colname="3" style="background-color:white;" colspan="1">(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510219.png" /> nodes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510220.png" />)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510221.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> <td colname="3" style="background-color:white;" colspan="1">( $  n $  nodes, $  n \geq 2 $ )</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  C _{n} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
</td> <td colname="3" style="background-color:white;" colspan="1">(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510222.png" /> nodes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510223.png" />)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510224.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> <td colname="3" style="background-color:white;" colspan="1">( $  n $  nodes, $  n \geq 3 $ )</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  D _{n} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
</td> <td colname="3" style="background-color:white;" colspan="1">(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510225.png" /> nodes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510226.png" />)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510227.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> <td colname="3" style="background-color:white;" colspan="1">( $  n $  nodes, $  n \geq 4 $ )</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _{6} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
</td> <td colname="3" style="background-color:white;" colspan="1">(6 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510228.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> <td colname="3" style="background-color:white;" colspan="1">(6 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _{7} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
</td> <td colname="3" style="background-color:white;" colspan="1">(7 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510229.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> <td colname="3" style="background-color:white;" colspan="1">(7 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _{8} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
</td> <td colname="3" style="background-color:white;" colspan="1">(8 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510230.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> <td colname="3" style="background-color:white;" colspan="1">(8 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  F _{4} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
</td> <td colname="3" style="background-color:white;" colspan="1">(4 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510231.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> <td colname="3" style="background-color:white;" colspan="1">(4 nodes)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  G _{2} $ </td> <td colname="2" style="background-color:white;" colspan="1">
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
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Figure: l058510a
 
Figure: l058510a
  
On the diagonal of the Cartan matrix all elements are equal to 2. If nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510232.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510233.png" /> are not directly linked, then the matrix entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510234.png" />. If two nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510235.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510236.png" /> are directly linked by a single edge, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510237.png" />. If two nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510238.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510239.png" /> are directly linked by a double, respectively triple, edge and the arrow points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510240.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510241.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510242.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510243.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510244.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510245.png" />.
+
On the diagonal of the Cartan matrix all elements are equal to 2. If nodes $  i $  and $  j $  are not directly linked, then the matrix entries $  a _{ji} = a _{ij} = 0 $ . If two nodes $  i $ , $  j $  are directly linked by a single edge, then $  a _{ij} = - 1 = a _{ji} $ . If two nodes $  i $ , $  j $  are directly linked by a double, respectively triple, edge and the arrow points from $  i $  to $  j $ , then $  a _{ij} = - 2 $ , $  a _{ji} = - 1 $ , respectively $  a _{ij} = - 3 $ , $  a _{ji} = - 1 $ .
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.B. Frenkel, V.G. Kac, "Basic representations of affine Lie algebras and dual resonance models" ''Invent. Math.'' , '''62''' (1980) pp. 23–66 {{MR|0595581}} {{ZBL|0493.17010}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966) {{MR|0215886}} {{ZBL|0144.02105}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.B. Frenkel, V.G. Kac, "Basic representations of affine Lie algebras and dual resonance models" ''Invent. Math.'' , '''62''' (1980) pp. 23–66 {{MR|0595581}} {{ZBL|0493.17010}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966) {{MR|0215886}} {{ZBL|0144.02105}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR></table>

Revision as of 17:35, 12 December 2019

A Lie algebra that has no non-zero solvable ideals (see Lie algebra, solvable). Henceforth finite-dimensional semi-simple Lie algebras over a field $ k $ of characteristic 0 are considered (for semi-simple Lie algebras over a field of non-zero characteristic see Lie algebra).

The fact that a finite-dimensional Lie algebra $ \mathfrak g $ is semi-simple is equivalent to any of the following conditions:

1) $ \mathfrak g $ does not contain non-zero Abelian ideals;

2) the Killing form of $ \mathfrak g $ is non-singular (Cartan's criterion);

3) $ \mathfrak g $ splits into the direct sum of non-Abelian simple ideals;

4) every finite-dimensional linear representation of $ \mathfrak g $ is completely reducible (in other words: every finite-dimensional $ \mathfrak g $ -module is semi-simple);

5) the one-dimensional cohomology of $ \mathfrak g $ with values in an arbitrary finite-dimensional $ \mathfrak g $ -module is trivial.

Any ideal and any quotient algebra of a semi-simple Lie algebra is also semi-simple. The decomposition of a semi-simple Lie algebra mentioned in condition 3) is unique. A special case of condition 5) is the following assertion: All derivations of a semi-simple Lie algebra are inner. The property of a Lie algebra of being semi-simple is preserved by both extensions and restrictions of the ground field.

Let $ \mathfrak g $ be a semi-simple Lie algebra over an algebraically closed field $ k $ . The adjoint representation maps $ \mathfrak g $ isomorphically onto the linear Lie algebra $ \mathop{\rm ad}\nolimits \ \mathfrak g $ , which is the Lie algebra of the algebraic group $ \mathop{\rm Aut}\nolimits \ \mathfrak g $ of all automorphisms of $ \mathfrak g $ and is therefore an algebraic Lie algebra (cf. Lie algebra, algebraic). An element $ X \in \mathfrak g $ is said to be semi-simple (nilpotent) if $ \mathop{\rm ad}\nolimits \ X $ is semi-simple (respectively, nilpotent). This property of an element $ X $ is preserved by any homomorphism of $ \mathfrak g $ into another semi-simple Lie algebra. The identity component $ ( \mathop{\rm Aut}\nolimits \ \mathfrak g ) ^{0} $ coincides with the group of inner automorphisms of $ \mathfrak g $ , that is, it is generated by the automorphisms of the form $ \mathop{\rm exp}\nolimits ( \mathop{\rm ad}\nolimits \ X ) $ , $ X \in \mathfrak g $ .

In the study of semi-simple Lie algebras over an algebraically closed field $ k $ an important role is played by the roots of a semi-simple Lie algebra, which are defined as follows. Let $ \mathfrak h $ be a Cartan subalgebra of $ \mathfrak g $ . For a non-zero linear function $ \alpha \in \mathfrak h ^ \ast $ , let $ \mathfrak g _ \alpha $ denote the linear subspace of $ \mathfrak g $ given by the condition $$ \mathfrak g _ \alpha = \{ {X \in \mathfrak g} : {[ H ,\ X ] = \alpha (H) X , H \in \mathfrak h} \} . $$ If $ \mathfrak g _ \alpha \neq 0 $ , then $ \alpha $ is called a root of $ \mathfrak g $ with respect to $ \mathfrak h $ . The set $ \Sigma $ of all non-zero roots is called the root system, or system of roots, of $ \mathfrak g $ . One has the root decomposition $$ \mathfrak g = \mathfrak h + \sum _ {\alpha \in \Sigma} \mathfrak g _ \alpha . $$ The root system and the root decomposition of a semi-simple Lie algebra have the following properties:

a) $ \Sigma $ generates $ \mathfrak h ^ \ast $ and is a reduced root system in the abstract sense (in the linear hull of $ \Sigma $ over the field of the real numbers). The system $ \Sigma $ is irreducible if and only if $ \mathfrak g $ is simple.

b) For any $ \alpha \in \Sigma $ , $$ \mathop{\rm dim}\nolimits \ \mathfrak g _ \alpha = \mathop{\rm dim}\nolimits \ [ \mathfrak g _ \alpha ,\ \mathfrak g _ {- \alpha} ] = 1 . $$ There is a unique element $ H _ \alpha \in [ \mathfrak g _ \alpha ,\ \mathfrak g _ {- \alpha} ] $ such that $ \alpha ( H _ \alpha ) = 2 $ .

c) For every non-zero $ X _ \alpha \in \mathfrak g _ \alpha $ there is a unique $ Y _ \alpha \in \mathfrak g _ {- \alpha} $ such that $ [ X _ \alpha ,\ Y _ \alpha ] = H _ \alpha $ , and $$ [ H _ \alpha ,\ X _ \alpha ] = 2 X _ \alpha \textrm{ and } [ H _ \alpha ,\ Y _ \alpha ] = - 2 Y _ \alpha . $$ Moreover, $$ \beta ( H _ \alpha ) = \frac{2 ( \alpha ,\ \beta )}{( \alpha ,\ \alpha )} , \alpha ,\ \beta \in \Sigma , $$ where $ ( \ ,\ ) $ is the scalar product induced by the Killing form.

d) If $ \alpha ,\ \beta \in \Sigma $ and $ \alpha + \beta \neq 0 $ , then $ \mathfrak g _ \alpha $ and $ \mathfrak g _ \beta $ are orthogonal with respect to the Killing form and $ [ \mathfrak g _ \alpha ,\ \mathfrak g _ \beta ] = \mathfrak g _ {\alpha + \beta} $ .

A basis $ \{ \alpha _{1} \dots \alpha _{n} \} $ of the root system $ \Sigma $ is also called a system of simple roots of the algebra $ \mathfrak g $ . Let $ \Sigma _{+} $ be the system of positive roots with respect to the given basis and let $ X _ {- \alpha} = Y _ \alpha $ $ ( \alpha \in \Sigma _{+} ) $ . Then the elements $$ H _ {\alpha _{1}} \dots H _ {\alpha _{k}} ,\ X _ \alpha ( \alpha \in \Sigma ) $$ form a basis of $ \mathfrak g $ , called a Cartan basis. On the other hand, the elements $$ X _ {\alpha _{i}} , X _ {- \alpha _{i}} ( i = 1 \dots n ) $$ form a system of generators of $ \mathfrak g $ , and the defining relations have the following form: $$ [ [ X _ {\alpha _{i}} ,\ X _ {- \alpha _{i}} ] ,\ X _ {\alpha _{j}} ] = n ( i ,\ j ) X _ {\alpha _{j}} , $$ $$ [ [ X _ {\alpha _{i}} ,\ X _ {- \alpha _{i}} ] ,\ X _ {- \alpha _{j}} ] = - n ( i ,\ j ) X _ {\alpha _{j}} , $$ $$ ( \mathop{\rm ad}\nolimits \ X _ {\alpha _{i}} ) ^ {1 - n ( i ,\ j )} X _ {\alpha _{j}} = 0 , $$ $$ ( \mathop{\rm ad}\nolimits \ X _ {- \alpha _{i}} ) ^ {1 - n ( i ,\ j )} X _ {- \alpha _{j}} = 0 . $$ Here $ i ,\ j = 1 \dots n $ and $$ n ( i ,\ j ) = \alpha _{j} ( H _{i} ) = \frac{2 ( \alpha _{i} ,\ \alpha _{j} )}{( \alpha _{j} ,\ \alpha _{j} )} . $$ Property d) implies that $$ [ X _ \alpha ,\ X _ \beta ] = \left \{ \begin{array}{ll} N _ {\alpha , \beta} X _ {\alpha + \beta} & \textrm{ if } \alpha + \beta \in \Sigma , \\ 0 & \textrm{ if } \alpha + \beta \notin \Sigma , \\ \end{array} \right .$$where $ N _ {\alpha , \beta} \in k $ . The elements $ X _ \alpha $ $ ( \alpha \in \Sigma _{+} ) $ can be chosen in such a way that $$ N _ {\alpha , \beta} = - N _ {- \alpha , - \beta} \textrm{ and } N _ {\alpha , \beta} = \pm ( p + 1 ) , $$ where $ p $ is the largest integer such that $ \beta - p \alpha \in \Sigma $ . The corresponding Cartan basis is called a Chevalley basis. The structure constants of $ \mathfrak g $ in this basis are integers, which makes it possible to associate with $ \mathfrak g $ Lie algebras and algebraic groups (see Chevalley group) over fields of arbitrary characteristic. If $ k = \mathbf C $ , then the linear hull over $ \mathbf R $ of the vectors $$ i H _ \alpha , X _ \alpha - X _ {- \alpha} , i ( X _ \alpha + X _ {- \alpha} ) \quad ( \alpha \in \Sigma _{+} ) $$ is a compact real form of $ \mathfrak g $ .

A semi-simple Lie algebra is defined up to an isomorphism by its Cartan subalgebra and the corresponding root system. More precisely, if $ \mathfrak g _{1} $ and $ \mathfrak g _{2} $ are semi-simple Lie algebras over $ k $ , $ \mathfrak h _{1} $ and $ \mathfrak h _{2} $ are their Cartan subalgebras and $ \Sigma _{1} $ and $ \Sigma _{2} $ are the corresponding root systems, then every isomorphism $ \mathfrak h _{1} \rightarrow \mathfrak h _{2} $ that induces an isomorphism of the root systems $ \Sigma _{1} $ and $ \Sigma _{2} $ can be extended to an isomorphism $ \mathfrak g _{1} \rightarrow \mathfrak g _{2} $ . On the other hand, any reduced root system can be realized as the root system of some semi-simple Lie algebra. Thus, the classification of semi-simple Lie algebras (respectively, simple non-Abelian Lie algebras) over an algebraically closed field $ k $ essentially coincides with the classification of reduced root systems (respectively, irreducible reduced root systems).

Simple Lie algebras that correspond to root systems of types $ A $ – $ D $ are said to be classical and have the following form.

Type $ A _{n} $ , $ n \geq 1 $ . $ \mathfrak g = \mathfrak s \mathfrak l ( n + 1 ,\ k ) $ , the algebra of linear transformations of the space $ k ^{n+1} $ with trace 0; $ \mathop{\rm dim}\nolimits \ \mathfrak g = n ( n + 2 ) $ .

Type $ B _{n} $ , $ n \geq 2 $ . $ \mathfrak g = \mathfrak s \mathfrak o ( 2 n + 1 ,\ k ) $ , the algebra of linear transformations of the space $ k ^{2n+1} $ that are skew-symmetric with respect to a given non-singular symmetric bilinear form; $ \mathop{\rm dim}\nolimits \ \mathfrak g = n ( 2 n + 1 ) $ .

Type $ C _{n} $ , $ n \geq 3 $ . $ \mathfrak g = \mathfrak s \mathfrak p ( n ,\ k ) $ , the algebra of linear transformations of the space $ k ^{2n} $ that are skew-symmetric with respect to a given non-singular skew-symmetric bilinear form; $ \mathop{\rm dim}\nolimits \ \mathfrak g = n ( 2 n + 1 ) $ .

Type $ D _{n} $ , $ n \geq 4 $ . $ \mathfrak g = \mathfrak s \mathfrak o ( 2 n ,\ k ) $ , the algebra of linear transformations of the space $ k ^{2n} $ that are skew-symmetric with respect to a given non-singular symmetric bilinear form; $ \mathop{\rm dim}\nolimits \ \mathfrak g = n ( 2 n - 1 ) $ .

The simple Lie algebras corresponding to the root systems of types $ E _{6} $ , $ E _{7} $ , $ E _{8} $ , $ F _{4} $ , $ G _{2} $ are called special, or exceptional (see Lie algebra, exceptional).

The Cartan matrix of a semi-simple Lie algebra over an algebraically closed field also determines this algebra uniquely up to an isomorphism. The Cartan matrices of the simple Lie algebras have the following form: $$ A _{n} : \left \| \begin{array}{rrrcr} 2 &-1 & 0 &\dots & 0 \\ -1 & 2 &-1 &\dots & 0 \\ 0 &-1 & 2 &\dots & 0 \\ . & . & . &\ddots & . \\ 0 & 0 & 0 &\dots &-1 \\ 0 & 0 & 0 &\dots & 2 \\ \end{array} \right \| , $$ $$ B _{n} : \left \| \begin{array}{rrrrrr} 2 &-1 & 0 &\dots & 0 & 0 \\ -1 & 2 &-1 &\dots & 0 & 0 \\ 0 &-1 & 2 &\dots & 0 & 0 \\ . & . & . &\ddots & . & . \\ 0 & 0 & 0 &\dots & 2 &-2 \\ 0 & 0 & 0 &\dots &-1 & 2 \\ \end{array} \right \| , $$ $$ C _{n} : \left \| \begin{array}{rrrrrrr} 2 &-1 & 0 &\dots & 0 & 0 \\ -1 & 2 &-1 &\dots & 0 & 0 \\ 0 &-1 & 2 &\dots & 0 & 0 \\ . & . & . &\cdots & . & . \\ 0 & 0 & 0 &\dots & 2 &-1 \\ 0 & 0 & 0 &\dots &-2 & 2 \\ \end{array} \right \| , $$ $$ D _{n} : \left \| \begin{array}{rrrrrrrr} 2 &-1 & 0 &\dots & 0 & 0 & 0 & 0 \\ -1 & 2 &-1 &\dots & 0 & 0 & 0 & 0 \\ 0 &-1 & 2 &\dots & 0 & 0 & 0 & 0 \\ . & . & . &\ddots & . & . & . & . \\ 0 & 0 & 0 &\dots & 2 &-1 & 0 & 0 \\ 0 & 0 & 0 &\dots &-1 & 2 &-1 &-1 \\ 0 & 0 & 0 &\dots & 0 &-1 & 2 & 0 \\ 0 & 0 & 0 &\dots & 0 &-1 & 0 & 2 \\ \end{array} \right \| , $$ $$ E _{6} : \left \| \begin{array}{rrrrrr} 2 & 0 &-1 & 0 & 0 & 0 \\ 0 & 2 & 0 &-1 & 0 & 0 \\ -1 & 0 & 2 &-1 & 0 & 0 \\ 0 &-1 &-1 & 2 &-1 & 0 \\ 0 & 0 & 0 &-1 & 2 &-1 \\ 0 & 0 & 0 & 0 &-1 & 2 \\ \end{array} \right \| , $$ $$ E _{7} : \left \| \begin{array}{rrrrrrr} 2 & 0 &-1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 &-1 & 0 & 0 & 0 \\ -1 & 0 & 2 &-1 & 0 & 0 & 0 \\ 0 &-1 &-1 & 2 &-1 & 0 & 0 \\ 0 & 0 & 0 &-1 & 2 &-1 & 0 \\ 0 & 0 & 0 & 0 &-1 & 2 &-1 \\ 0 & 0 & 0 & 0 & 0 &-1 & 2 \\ \end{array} \right \| , $$ $$ E _{8} : \left \| \begin{array}{rrrrrrrr} 2 & 0 &-1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 &-1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 2 &-1 & 0 & 0 & 0 & 0 \\ 0 &-1 &-1 & 2 &-1 & 0 & 0 & 0 \\ 0 & 0 & 0 &-1 & 2 &-1 & 0 & 0 \\ 0 & 0 & 0 & 0 &-1 & 2 &-1 & 0 \\ 0 & 0 & 0 & 0 & 0 &-1 & 2 &-1 \\ 0 & 0 & 0 & 0 & 0 & 0 &-1 & 2 \\ \end{array} \right \| , $$ $$ F _{4} : \left \| \begin{array}{rrrr} 2 &-1 & 0 & 0 \\ -1 & 2 &-2 & 0 \\ 0 &-1 & 2 &-1 \\ 0 & 0 &-1 & 2 \\ \end{array} \right \| , G _{2} : \left \| \begin{array}{rr} 2 &-1 \\ -3 & 2 \\ \end{array} \right \| . $$ The classification of split semi-simple Lie algebras over an arbitrary field $ k $ of characteristic zero (a semi-simple Lie algebra $ \mathfrak g $ is said to be split if it has a Cartan subalgebra $ \mathfrak h \subset \mathfrak g $ such that all characteristic roots of the operators $ \mathop{\rm ad}\nolimits \ X $ , $ X \in \mathfrak h $ , lie in $ k $ ) goes in the same way as in the case of an algebraically closed field. Namely, to every irreducible reduced root system corresponds a unique split semi-simple Lie algebra. In particular, split semi-simple Lie algebras of types $ A $ – $ D $ have the form stated above, except that in the cases $ B $ and $ D $ one must consider non-singular symmetric bilinear forms with Witt index $ n $ .

The problem of classifying arbitrary semi-simple Lie algebras over $ k $ reduces to the following problem: To list, up to an isomorphism, all $ k $ -forms $ \mathfrak g _{0} \subset \mathfrak g $ , that is, all $ k $ -subalgebras $ \mathfrak g _{0} \subset \mathfrak g $ such that $ \mathfrak g = \mathfrak g _{0} \otimes _{k} K $ . Here $ K $ is an algebraically closed extension of $ k $ and $ \mathfrak g $ is a given semi-simple Lie algebra over $ K $ . The solution of this problem can also be obtained in terms of root systems (see Form of an algebraic group; Form of an (algebraic) structure). When $ \mathfrak g $ is a classical simple Lie algebra over $ k $ (other than $ D _{4} $ ), there is another method of classifying $ k $ -forms in $ \mathfrak g $ , based on an examination of simple associative algebras (see [3]).

When $ k = \mathbf R $ the classification of semi-simple Lie algebras goes as follows (see [6], [7]). Every simple non-Abelian Lie algebra over $ \mathbf R $ is either a simple Lie algebra over $ \mathbf C $ (regarded as an algebra over $ \mathbf R $ ), or the real form of a simple Lie algebra over $ \mathbf C $ . The classification of real forms $ \mathfrak g _{0} $ in a simple classical Lie algebra $ \mathfrak g $ over $ \mathbf C $ is as follows:

I) Type $ A _{n} $ : $ \mathfrak g = \mathfrak s \mathfrak l ( n + 1 ,\ \mathbf C ) $ , $ n \geq 1 $ . $ A _{I} $ : $ \mathfrak g _{0} = \mathfrak s \mathfrak l ( n + 1 ,\ \mathbf R ) $ . $ A _{II} $ : $ n + 1 = 2 m $ is even, $ \mathfrak g _{0} = \mathfrak s \mathfrak u ^ \ast ( 2 n ) $ , the subalgebra of elements of $ \mathfrak s \mathfrak l ( 2 m ,\ \mathbf C ) $ that preserve a certain quaternion structure. $ A _{III} $ : $ \mathfrak g _{0} = \mathfrak s \mathfrak u ( p ,\ n + 1 - p ) $ , the subalgebra of elements of $ \mathfrak s \mathfrak l ( n + 1 ,\ \mathbf C ) $ that are skew-symmetric with respect to a non-singular Hermitian form of positive index $ p $ , $ 0 \leq p \leq ( n + 1 ) / 2 . $

II) Type $ B _{n} $ : $ \mathfrak g = \mathfrak s \mathfrak o ( 2 n + 1 ,\ \mathbf C ) $ , $ n \geq 2 $ . $ B _{I} $ : $ \mathfrak g _{0} = \mathfrak s \mathfrak o ( p ,\ 2 n + 1 - p ) $ , the algebra of a linear transformations of the space $ \mathbf R ^{2n+1} $ that are skew-symmetric with respect to a non-singular symmetric bilinear form of positive index $ p $ , $ 0 \leq p \leq n $ .

III) Type $ C _{n} $ : $ \mathfrak g = \mathfrak s \mathfrak p (n,\ \mathbf C ) $ , $ n \geq 3 $ . $ C _{I} $ : $ \mathfrak g _{0} = \mathfrak s \mathfrak p ( n ,\ \mathbf R ) $ , the algebra of linear transformations of the space $ \mathbf R ^{2n} $ that are skew-symmetric with respect to a non-singular skew-symmetric bilinear form. $ C _{II} $ : $ \mathfrak g _{0} = \mathfrak s \mathfrak p ( p ,\ n - p ) $ , $ 0 \leq p \leq n / 2 $ , the subalgebra of $ \mathfrak s \mathfrak u ( 2 p ,\ 2 ( n - p ) ) $ consisting of transformations that preserve a certain quaternion structure.

IV) Type $ D _{n} $ : $ \mathfrak g = \mathfrak s \mathfrak o ( 2 n ,\ \mathbf C ) $ , $ n \geq 4 $ . $ D _{I} $ : $ \mathfrak g _{0} = \mathfrak s \mathfrak o ( p ,\ 2 n - p ) $ , the algebra of linear transformations of the space $ \mathbf R ^{2n} $ that are skew-symmetric with respect to a non-singular bilinear symmetric form of positive index $ p $ , $ 0 \leq p \leq n $ . $ D _{III} $ : $ \mathfrak g _{0} = \mathfrak s \mathfrak o ^ \ast ( 2 n ,\ \mathbf C ) $ , the subalgebra of $ \mathfrak s \mathfrak o (2n ,\ \mathbf C ) $ consisting of transformations that preserve a certain quaternion structure.

Semi-simple Lie algebras over the field $ \mathbf C $ were first considered in papers by W. Killing , who gave a classification of them, although in his proofs there were gaps, which were filled by E. Cartan [2]. In the papers of Killing and Cartan the roots of a Lie algebra appeared as the characteristic roots of the operator $ \mathop{\rm ad}\nolimits \ X $ . Cartan also gave a classification of real semi-simple Lie algebras by establishing a deep connection between these algebras and globally symmetric Riemannian spaces (cf. Globally symmetric Riemannian space).

References

[1a] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen I" Math. Ann. , 31 (1888) pp. 252–290 Zbl 20.0368.03
[1b] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen II" Math. Ann. , 33 (1889) pp. 1–48 Zbl 20.0368.03
[1c] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen III" Math. Ann. , 34 (1889) pp. 57–122 Zbl 21.0376.01
[1d] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen IV" Math. Ann. , 36 (1890) pp. 161–189 MR1510618
[2] E. Cartan, "Sur la structure des groupes de transformations finis et continues" , Oeuvres complètes , 1 , Gauthier-Villars (1952) pp. 137–287
[3] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[4] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[5] R.G. Steinberg, "Lectures on Chevalley groups" , Yale Univ. Press (1967) MR0476871 MR0466335 Zbl 0307.22001 Zbl 1196.22001
[6] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038
[7] S. Araki, "On root systems and an infinitesimal classification of irreducible symmetric spaces" Osaka J. Math. , 13 (1962) pp. 1–34 MR0153782 Zbl 0123.03002


Comments

The defining relations, mentioned above, $ ( \mathop{\rm ad}\nolimits \ X _ {\alpha _{i}} ) ^ {1 - n ( i , j )} ( X _ {\alpha _{j}} ) = 0 $ , are known as the Serre relations.

It is customary to encode the information contained in the Cartan matrices $ A _{n} $ – $ G _{2} $ by means of the so-called Dynkin diagrams.'

<tbody> </tbody>
$ A _{n} $

( $ n $ nodes)
$ B _{n} $

( $ n $ nodes, $ n \geq 2 $ )
$ C _{n} $

( $ n $ nodes, $ n \geq 3 $ )
$ D _{n} $

( $ n $ nodes, $ n \geq 4 $ )
$ E _{6} $

(6 nodes)
$ E _{7} $

(7 nodes)
$ E _{8} $

(8 nodes)
$ F _{4} $

(4 nodes)
$ G _{2} $

(2 nodes)

The rules for recovering the Cartan matrix from the corresponding Dynkin diagram (also called Dynkin graph occasionally) are as follows. Number the vertices, e.g.,

Figure: l058510a

On the diagonal of the Cartan matrix all elements are equal to 2. If nodes $ i $ and $ j $ are not directly linked, then the matrix entries $ a _{ji} = a _{ij} = 0 $ . If two nodes $ i $ , $ j $ are directly linked by a single edge, then $ a _{ij} = - 1 = a _{ji} $ . If two nodes $ i $ , $ j $ are directly linked by a double, respectively triple, edge and the arrow points from $ i $ to $ j $ , then $ a _{ij} = - 2 $ , $ a _{ji} = - 1 $ , respectively $ a _{ij} = - 3 $ , $ a _{ji} = - 1 $ .

References

[a1] I.B. Frenkel, V.G. Kac, "Basic representations of affine Lie algebras and dual resonance models" Invent. Math. , 62 (1980) pp. 23–66 MR0595581 Zbl 0493.17010
[a2] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[a3] J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966) MR0215886 Zbl 0144.02105
[a4] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 MR0323842 Zbl 0254.17004
How to Cite This Entry:
Lie algebra, semi-simple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_semi-simple&oldid=21885
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article