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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  $  A _{n} $ – $  G _{2} $  by means of the so-called Dynkin diagrams.
  
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">
+
<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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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.,
 
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.,
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l058510a.gif" />
+
[[File:Dynkin E8.svg|center|200px|Dynkin diagram E8]]
 
 
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>
 +
[[Category:Nonassociative rings and algebras]]

Latest revision as of 19:24, 26 March 2023

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.,

Dynkin diagram E8

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