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A [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584301.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584302.png" /> that is graded by means of an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584303.png" />, that is, which splits into a direct sum of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584305.png" />, in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584306.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584307.png" /> is an [[Ordered group|ordered group]], then for every filtered Lie algebra (cf. [[Filtered algebra|Filtered algebra]]) the [[Graded algebra|graded algebra]] associated with it is a graded Lie algebra.
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Graded Lie algebras play an important role in the classification of simple finite-dimensional Lie algebras, Jordan algebras and their generalizations, and primitive pseudo-groups of transformations (see [[#References|[3]]], [[#References|[4]]]). For any semi-simple real Lie algebra its [[Cartan decomposition|Cartan decomposition]] can be regarded as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584308.png" />-grading. The local classification of symmetric Riemannian spaces reduces to the classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l0584309.png" />-graded simple complex Lie algebras [[#References|[6]]].
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A [[Lie algebra|Lie algebra]]  $  \mathfrak g $
 +
over a field  $  K $
 +
that is graded by means of an Abelian group  $  A $,
 +
that is, which splits into a direct sum of subspaces  $  \mathfrak g _  \alpha  $,
 +
$  \alpha \in A $,
 +
in such a way that  $  [ \mathfrak g _  \alpha  , \mathfrak g _  \beta  ] \subseteq \mathfrak g _ {\alpha + \beta }  $.
 +
If  $  A $
 +
is an [[Ordered group|ordered group]], then for every filtered Lie algebra (cf. [[Filtered algebra|Filtered algebra]]) the [[Graded algebra|graded algebra]] associated with it is a graded Lie algebra.
 +
 
 +
Graded Lie algebras play an important role in the classification of simple finite-dimensional Lie algebras, Jordan algebras and their generalizations, and primitive pseudo-groups of transformations (see [[#References|[3]]], [[#References|[4]]]). For any semi-simple real Lie algebra its [[Cartan decomposition|Cartan decomposition]] can be regarded as a $  \mathbf Z _ {2} $-
 +
grading. The local classification of symmetric Riemannian spaces reduces to the classification of $  \mathbf Z _ {2} $-
 +
graded simple complex Lie algebras [[#References|[6]]].
  
 
==Some constructions of graded Lie algebras.==
 
==Some constructions of graded Lie algebras.==
  
 +
1) Let  $  U $
 +
be an associative algebra (cf. [[Associative rings and algebras|Associative rings and algebras]]) endowed with an increasing filtration  $  ( U _ {k} : k \in \mathbf Z ) $,
 +
suppose that  $  [ U _ {k} , U _ {l} ] \subset  U _ {k+l-d} $,
 +
where  $  d $
 +
is a fixed natural number, and let  $  \mathfrak u _ {k} = U _ {k+d} / U _ {k+d-1} $.
 +
Then the commutation operation in  $  U $
 +
induces in the space  $  \mathfrak u = \sum _ {k = - \infty }  ^  \infty  \mathfrak u _ {k} $
 +
the structure of a  $  \mathbf Z $-
 +
graded Lie algebra. In this way one can obtain some Lie algebras of functions with the [[Poisson brackets|Poisson brackets]] as commutator. In the next two examples,  $  U _ {k} = U _ {1}  ^ {k} $
 +
for  $  k > 0 $
 +
and  $  U _ {k} = 0 $
 +
for  $  k < 0 $.
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843010.png" /> be an associative algebra (cf. [[Associative rings and algebras|Associative rings and algebras]]) endowed with an increasing filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843011.png" />, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843013.png" /> is a fixed natural number, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843014.png" />. Then the commutation operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843015.png" /> induces in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843016.png" /> the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843017.png" />-graded Lie algebra. In this way one can obtain some Lie algebras of functions with the [[Poisson brackets|Poisson brackets]] as commutator. In the next two examples, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843020.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843021.png" />.
+
a) Let $  U $
 
+
be the algebra of linear differential operators with polynomial coefficients and let $  U _ {1} $
a) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843022.png" /> be the algebra of linear differential operators with polynomial coefficients and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843023.png" /> be the subspace spanned by its generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843026.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843028.png" /> is the Lie algebra of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843030.png" /> with the usual Poisson brackets.
+
be the subspace spanned by its generators $  p _ {i} = \partial  / {\partial  x _ {i} } $,  
 +
$  q _ {i} = x _ {i} $,
 +
$  i = 1 \dots m $.  
 +
Then  $  [ U _ {k} , U _ {l} ] \subset  U _ {k+l-2} $
 +
and $  \mathfrak u $
 +
is the Lie algebra of polynomials in $  p _ {i} $
 +
and $  q _ {i} $
 +
with the usual Poisson brackets.
  
b) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843031.png" /> be the [[Universal enveloping algebra|universal enveloping algebra]] of a finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843032.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843033.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843035.png" /> is canonically isomorphic (as a vector space) to the symmetric algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843036.png" />, that is, to the algebra of polynomials on the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843037.png" /> (the Poincaré–Birkhoff–Witt theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843038.png" /> is the Lie algebra of a connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843039.png" />, then the commutator of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843040.png" /> can be interpreted either as the Poisson brackets for the corresponding left-invariant functions on the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843041.png" />, or as the Poisson brackets on each orbit of the co-adjoint representation, defined by means of the standard symplectic structure on these orbits.
+
b) Let $  U $
 +
be the [[Universal enveloping algebra|universal enveloping algebra]] of a finite-dimensional Lie algebra $  \mathfrak g $
 +
and let $  U _ {1} = \mathfrak g $.  
 +
Then  $  [ U _ {k} , U _ {l} ] \subset  U _ {k+l-1} $
 +
and $  \mathfrak u $
 +
is canonically isomorphic (as a vector space) to the symmetric algebra over $  \mathfrak g $,  
 +
that is, to the algebra of polynomials on the dual space $  \mathfrak g  ^ {*} $(
 +
the Poincaré–Birkhoff–Witt theorem). If $  \mathfrak g $
 +
is the Lie algebra of a connected Lie group $  G $,  
 +
then the commutator of elements of $  \mathfrak u $
 +
can be interpreted either as the Poisson brackets for the corresponding left-invariant functions on the cotangent bundle $  T  ^ {*} G $,  
 +
or as the Poisson brackets on each orbit of the co-adjoint representation, defined by means of the standard symplectic structure on these orbits.
  
2) Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843042.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843043.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843044.png" />-dimensional vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843045.png" /> endowed with a non-singular quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843046.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843047.png" /> be an orthogonal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843048.png" />. The decomposition of the [[Clifford algebra|Clifford algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843049.png" /> into the sum of one-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843051.png" />, is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843052.png" />-grading of it. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843053.png" /> the elements of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843054.png" /> with zero trace form a simple graded Lie algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843056.png" />; its grading has a high degree of symmetry; in particular, all graded subspaces are equivalent. Similar gradings (by means of various finite groups) exist for other simple Lie algebras [[#References|[1]]].
+
2) Suppose that $  \mathop{\rm char}  k \neq 2 $
 +
and that $  E $
 +
is an $  n $-
 +
dimensional vector space over $  k $
 +
endowed with a non-singular quadratic form $  Q $;  
 +
let $  e _ {1} \dots e _ {n} $
 +
be an orthogonal basis of $  E $.  
 +
The decomposition of the [[Clifford algebra|Clifford algebra]] $  C (Q) $
 +
into the sum of one-dimensional subspaces $  \langle  e _ {i _ {1}  } \dots e _ {i _ {k}  } \rangle $,
 +
$  i _ {1} < \dots < i _ {k} $,  
 +
is a $  \mathbf Z _ {2}  ^ {n} $-
 +
grading of it. For $  n = 2m $
 +
the elements of the algebra $  C (Q) $
 +
with zero trace form a simple graded Lie algebra of type $  A _ {N} $,  
 +
$  N = 2  ^ {m} - 1 $;  
 +
its grading has a high degree of symmetry; in particular, all graded subspaces are equivalent. Similar gradings (by means of various finite groups) exist for other simple Lie algebras [[#References|[1]]].
  
3) To every Lie [[Pseudo-group|pseudo-group]] of transformations corresponds a Lie algebra of vector fields. The germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843057.png" /> of this Lie algebra at any point has a natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843058.png" />-filtration
+
3) To every Lie [[Pseudo-group|pseudo-group]] of transformations corresponds a Lie algebra of vector fields. The germ l $
 +
of this Lie algebra at any point has a natural $  \mathbf Z $-
 +
filtration
  
<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/l058430/l05843059.png" /></td> </tr></table>
+
$$
 +
= l _ {-1}  \supset  l _ {0}  \supset  l _ {1}  \supset \dots ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843060.png" /> contains the germs of those vector fields whose coordinates can be expanded in power series without terms of degree less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843061.png" />. The associated graded Lie algebra can be interpreted as a Lie algebra of polynomial vector fields.
+
where l _ {k} $
 +
contains the germs of those vector fields whose coordinates can be expanded in power series without terms of degree less than $  k + 1 $.  
 +
The associated graded Lie algebra can be interpreted as a Lie algebra of polynomial vector fields.
  
 
==The classification of simple graded Lie algebras.==
 
==The classification of simple graded Lie algebras.==
 
To simple primitive Lie pseudo-groups correspond the following four series of simple infinite-dimensional graded Lie algebras (see [[#References|[5]]]):
 
To simple primitive Lie pseudo-groups correspond the following four series of simple infinite-dimensional graded Lie algebras (see [[#References|[5]]]):
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843062.png" />, the Lie algebra of all polynomial vector fields in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843063.png" />-dimensional affine space;
+
$  W _ {n} $,  
 +
the Lie algebra of all polynomial vector fields in the $  n $-
 +
dimensional affine space;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843064.png" />, its subalgebra consisting of vector fields with zero divergence;
+
$  S _ {n} $,  
 +
its subalgebra consisting of vector fields with zero divergence;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843065.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843066.png" />, the subalgebra consisting of vector fields that annihilate the differential form
+
$  H _ {n} $,  
 +
where $  n = 2 m $,  
 +
the subalgebra consisting of vector fields that annihilate the differential 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/l058430/l05843067.png" /></td> </tr></table>
+
$$
 +
\sum _ { i=1 } ^ { m }
 +
d x _ {i} \wedge d x _ {m+i}  $$
  
 
(Hamiltonian vector fields);
 
(Hamiltonian vector fields);
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843068.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843069.png" />, the subalgebra consisting of vector fields that multiply the differential form
+
$  K _ {n} $,  
 +
where $  n = 2 m + 1 $,  
 +
the subalgebra consisting of vector fields that multiply the differential 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/l058430/l05843070.png" /></td> </tr></table>
+
$$
 +
\sum _ { i=1 } ^ { m }
 +
( x _ {m+i}  d x _ {i} - x _ {i}  d x _ {m+i} ) + d x _ {n}  $$
  
 
by a function.
 
by a function.
  
Over fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843071.png" /> one can define simple finite-dimensional graded Lie algebras analogous to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843074.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843075.png" /> (see [[#References|[5]]]).
+
Over fields of characteristic $  p > 0 $
 +
one can define simple finite-dimensional graded Lie algebras analogous to $  W _ {n} $,  
 +
$  S _ {n} $,  
 +
$  H _ {n} $,  
 +
and $  K _ {n} $(
 +
see [[#References|[5]]]).
  
Simple graded Lie algebras of another type are obtained in the following way [[#References|[4]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843076.png" /> be the Lie algebra defined by means of an indecomposable Cartan matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843078.png" /> (from now on the notation of the article [[Cartan matrix|Cartan matrix]] is used). The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843079.png" /> is endowed with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843080.png" />-grading so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843083.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843084.png" /> is the row <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843085.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843086.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843087.png" />-th place. Elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843088.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843089.png" /> are called roots, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843090.png" /> are called simple roots. Any root is a linear combination of simple roots with integer coefficients of the same sign and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843091.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843092.png" />. The quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843093.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843094.png" /> with respect to its centre, which lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843095.png" />, is simple as a graded algebra, that is, it does not have non-trivial graded ideals.
+
Simple graded Lie algebras of another type are obtained in the following way [[#References|[4]]]. Let $  \mathfrak g = \mathfrak g (A) $
 +
be the Lie algebra defined by means of an indecomposable Cartan matrix $  A = \| a _ {ij} \| $,
 +
$  i , j = 1 \dots n $(
 +
from now on the notation of the article [[Cartan matrix|Cartan matrix]] is used). The algebra $  \mathfrak g $
 +
is endowed with a $  \mathbf Z  ^ {k} $-
 +
grading so that $  h _ {i} \in \mathfrak g _ {0} $,  
 +
$  e _ {i} \in \mathfrak g _ {\alpha _ {i}  } $,  
 +
$  f _ {i} \in \mathfrak g _ {- \alpha _ {i}  } $,  
 +
where $  \alpha _ {i} $
 +
is the row $  ( 0 \dots 1 \dots 0 ) $
 +
with $  1 $
 +
in the $  i $-
 +
th place. Elements $  \alpha \in \mathbf Z  ^ {n} $
 +
for which $  \mathfrak g _  \alpha  \neq 0 $
 +
are called roots, and the $  \alpha _ {i} $
 +
are called simple roots. Any root is a linear combination of simple roots with integer coefficients of the same sign and $  \mathop{\rm dim}  \mathfrak g _  \alpha  < \infty $
 +
for any $  \alpha \in \mathbf Z  ^ {n} $.  
 +
The quotient algebra $  {\mathfrak g }  ^  \prime  (A) $
 +
of $  \mathfrak g $
 +
with respect to its centre, which lies in $  \mathfrak g _ {0} $,  
 +
is simple as a graded algebra, that is, it does not have non-trivial graded ideals.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843096.png" /> be the totality of linear combinations of rows of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843097.png" /> with positive coefficients. Then one of the following cases holds:
+
Let $  R $
 +
be the totality of linear combinations of rows of the matrix $  A $
 +
with positive coefficients. Then one of the following cases holds:
  
(P) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843098.png" /> contains a row all elements of which are positive;
+
(P) $  R $
 +
contains a row all elements of which are positive;
  
(Z) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l05843099.png" /> contains a zero row;
+
(Z) $  R $
 +
contains a zero row;
  
(N) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430100.png" /> contains a row all elements of which are negative.
+
(N) $  R $
 +
contains a row all elements of which are negative.
  
In the case (P), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430101.png" /> is a simple finite-dimensional Lie algebra. In the case (N), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430102.png" /> is a simple infinite-dimensional Lie algebra. In the case (Z), the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430103.png" /> is simple only as a graded algebra. It can be converted in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430104.png" />-algebra so that: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430106.png" /> is a row of positive numbers; and b) the quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430107.png" /> is a simple finite-dimensional Lie algebra. The greatest common divisor of all components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430108.png" /> of the row <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430109.png" />, which is equal to 1, 2 or 3, is called the index of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430110.png" />.
+
In the case (P), $  \mathfrak g (A) = \mathfrak g  ^  \prime  (A) $
 +
is a simple finite-dimensional Lie algebra. In the case (N), $  \mathfrak g (A) $
 +
is a simple infinite-dimensional Lie algebra. In the case (Z), the algebra $  \mathfrak g  ^  \prime  = \mathfrak g  ^  \prime  (A) $
 +
is simple only as a graded algebra. It can be converted in a $  K [ u , u  ^ {-1} ] $-
 +
algebra so that: a) $  u \mathfrak g _  \alpha  ^  \prime  = \mathfrak g _ {\alpha + \nu }  ^  \prime  $,  
 +
where $  \nu $
 +
is a row of positive numbers; and b) the quotient algebra $  \mathfrak g  ^  \prime  / ( 1 - u ) \mathfrak g  ^  \prime  = \overline{ {\mathfrak g }}\; $
 +
is a simple finite-dimensional Lie algebra. The greatest common divisor of all components $  \nu _ {i} $
 +
of the row $  \nu $,  
 +
which is equal to 1, 2 or 3, is called the index of the algebra $  \mathfrak g  ^  \prime  $.
  
The following table is a list of all simple graded Lie algebras with Cartan matrix of type (Z). Here the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430111.png" /> is denoted by the same symbol as the associated simple finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430112.png" />, but with the addition of its index, given in brackets.
+
The following table is a list of all simple graded Lie algebras with Cartan matrix of type (Z). Here the algebra $  \mathfrak g  ^  \prime  $
 +
is denoted by the same symbol as the associated simple finite-dimensional Lie algebra $  \overline{ {\mathfrak g }}\; $,  
 +
but with the addition of its index, given in brackets.
  
The diagram of simple roots describes the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430113.png" />. Its vertices correspond to the simple roots; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430114.png" />-th and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430115.png" />-th vertices are joined by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430116.png" />-multiple edge, directed from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430117.png" />-th vertex to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430118.png" />-th if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430119.png" />, and undirected if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430120.png" />. Above the vertices stand the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430121.png" />.''''''<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">Notation</td> <td colname="2" style="background-color:white;" colspan="1">Diagram of simple roots</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/l058430/l058430122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430123.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
The diagram of simple roots describes the matrix $  A $.  
 +
Its vertices correspond to the simple roots; the $  i $-
 +
th and $  j $-
 +
th vertices are joined by an $  ( a _ {ij} a _ {ji} ) $-
 +
multiple edge, directed from the $  i $-
 +
th vertex to the $  j $-
 +
th if $  | a _ {ij} | > | a _ {ji} | $,  
 +
and undirected if $  | a _ {ij} | = | a _ {ji} | $.  
 +
Above the vertices stand the numbers $  \nu _ {i} $.
 +
<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">Notation</td> <td colname="2" style="background-color:white;" colspan="1">Diagram of simple roots</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {n-1}  ^ {(1)} $,  
 +
$  n \geq  3 $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430124.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {1}  ^ {(1)} $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430126.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {2n-2}  ^ {(2)} $,  
 +
$  n \geq  3 $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430127.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {2}  ^ {(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/" />
  
</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/l058430/l058430128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430129.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {2n-3}  ^ {(2)} $,  
 +
$  n \geq  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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430130.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430131.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  B _ {n-1}  ^ {(1)} $,  
 +
$  n \geq  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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430132.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430133.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  C _ {n-1}  ^ {(1)} $,  
 +
$  n \geq  3 $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430134.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430135.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  D _ {n-1}  ^ {(1)} $,  
 +
$  n \geq  5 $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430136.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430137.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  D _ {n}  ^ {(2)} $,  
 +
$  n \geq  3 $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430138.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  D _ {4}  ^ {(3)} $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430139.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _ {6}  ^ {(1)} $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430140.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _ {6}  ^ {(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/" />
  
</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/l058430/l058430141.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _ {7}  ^ {(1)} $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430142.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _ {8}  ^ {(1)} $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430143.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  F _ {4}  ^ {(1)} $
 +
</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> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430144.png" /></td> <td colname="2" style="background-color:white;" colspan="1">
+
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  G _ {2}  ^ {(1)} $
 +
</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/" />
Line 123: Line 279:
 
</td></tr> </table>
 
</td></tr> </table>
  
By means of graded Lie algebras with Cartan matrix of type (Z) one can classify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430145.png" />-graded simple finite-dimensional Lie algebras (see [[#References|[4]]], [[#References|[2]]]). Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430146.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430147.png" /> satisfies condition (Z), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430148.png" /> be a homomorphism such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430149.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430150.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430151.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430152.png" /> is mapped isomorphically onto the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430153.png" />, which depends only on the residue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430154.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430155.png" />, and the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430156.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430157.png" />-grading of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430158.png" />. If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430159.png" /> is algebraically closed, then by the method described one obtains, without repetition, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430160.png" />-graded simple finite-dimensional Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430161.png" />. The index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430162.png" /> is equal to the order of the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430163.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430164.png" />, of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430165.png" /> modulo the group of inner automorphisms.
+
By means of graded Lie algebras with Cartan matrix of type (Z) one can classify $  \mathbf Z _ {m} $-
 +
graded simple finite-dimensional Lie algebras (see [[#References|[4]]], [[#References|[2]]]). Namely, let $  \mathfrak g  ^  \prime  = \mathfrak g  ^  \prime  (A) $,  
 +
where $  A $
 +
satisfies condition (Z), and let $  p : \mathbf Z  ^ {n} \rightarrow \mathbf Z $
 +
be a homomorphism such that $  p ( \alpha _ {i} ) \geq  0 $
 +
and $  p ( \nu ) = m $.  
 +
Then $  \mathfrak g _ {k}  ^  \prime  = \sum _ {p ( \alpha ) = k }  \mathfrak g _  \alpha  ^  \prime  $
 +
for any $  k \in \mathbf Z $
 +
is mapped isomorphically onto the subspace $  \overline{ {\mathfrak g }}\; _ {k} \subset  \overline{ {\mathfrak g }}\; $,  
 +
which depends only on the residue of $  k $
 +
modulo $  m $,  
 +
and the decomposition $  \overline{ {\mathfrak g }}\; = \sum _ {k=0}  ^ {m-1} \overline{ {\mathfrak g }}\; _ {k} $
 +
is a $  \mathbf Z _ {m} $-
 +
grading of $  \overline{ {\mathfrak g }}\; $.  
 +
If the field $  K $
 +
is algebraically closed, then by the method described one obtains, without repetition, all $  \mathbf Z _ {m} $-
 +
graded simple finite-dimensional Lie algebras over $  K $.  
 +
The index of $  \mathfrak g  ^  \prime  $
 +
is equal to the order of the automorphism $  \theta : x \mapsto (  \mathop{\rm exp} ( {2 \pi i k } / m )) x $,  
 +
$  x \in \mathfrak g _ {k} $,  
 +
of the algebra $  \overline{ {\mathfrak g }}\; $
 +
modulo the group of inner automorphisms.
  
There is a classification of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430166.png" />-graded Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430167.png" /> satisfying the conditions: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430168.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430170.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430171.png" /> is generated by the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430172.png" />; and c) the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430173.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430174.png" /> is irreducible. In this case either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430175.png" /> is finite-dimensional or it is one of the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430176.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430177.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430178.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430179.png" />, or it is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430180.png" /> defined by a Cartan matrix of type (Z), endowed with a suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430181.png" />-grading [[#References|[4]]].
+
There is a classification of simple $  \mathbf Z $-
 +
graded Lie algebras $  \mathfrak g = \sum _ {k = - \infty }  ^  \infty  \mathfrak g _ {k} $
 +
satisfying the conditions: a) $  \mathop{\rm dim}  \mathfrak g _ {k} \leq  C  | k |  ^ {N} $
 +
for some $  C $
 +
and $  N $;  
 +
b) $  \mathfrak g $
 +
is generated by the subspace $  \mathfrak g _ {-1} + \mathfrak g _ {0} + \mathfrak g _ {1} $;  
 +
and c) the representation of $  \mathfrak g _ {0} $
 +
on $  \mathfrak g _ {-1} $
 +
is irreducible. In this case either $  \mathfrak g $
 +
is finite-dimensional or it is one of the algebras $  W _ {n} $,  
 +
$  S _ {n} $,  
 +
$  H _ {n} $,  
 +
$  K _ {n} $,  
 +
or it is the algebra $  \mathfrak g  ^  \prime  (A) $
 +
defined by a Cartan matrix of type (Z), endowed with a suitable $  \mathbf Z $-
 +
grading [[#References|[4]]].
  
A Lie [[Superalgebra|superalgebra]] is sometimes called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430182.png" />-graded Lie algebra.
+
A Lie [[Superalgebra|superalgebra]] is sometimes called a $  \mathbf Z _ {2} $-
 +
graded Lie algebra.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Alekseevskii,  "Finite commutative Jordan subgroups of complex simple Lie groups"  ''Funct. Anal. Appl.'' , '''8''' :  4  (1974)  pp. 277–279  ''Funktsional. Anal. Prilozhen.'' , '''8''' :  4  (1974)  pp. 1–4</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.B. Vinberg,  "The Weyl group of a graded Lie algebra"  ''Math. USSR Izv.'' , '''10'''  (1976)  pp. 436–496  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''40''' :  3  (1976)  pp. 488–526</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.L. Kantor,  "Certain generalizations of Jordan algebras"  ''Trudy Sem. Vektor. Tenzor. Anal.'' , '''16'''  (1972)  pp. 407–499  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.G. Kac,  "Simple irreducible graded Lie algebras of finite growth"  ''Math. USSR Izv.'' , '''2'''  (1968)  pp. 1271–1312  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''32''' :  6  (1968)  pp. 1323–1367</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.I. Kostrikin,  I.R. Shafarevich,  "Graded Lie algebras of finite characteristic"  ''Math. USSR Izv.'' , '''3'''  (1969)  pp. 237–304  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''33''' :  2  (1969)  pp. 252–322</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Alekseevskii,  "Finite commutative Jordan subgroups of complex simple Lie groups"  ''Funct. Anal. Appl.'' , '''8''' :  4  (1974)  pp. 277–279  ''Funktsional. Anal. Prilozhen.'' , '''8''' :  4  (1974)  pp. 1–4</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.B. Vinberg,  "The Weyl group of a graded Lie algebra"  ''Math. USSR Izv.'' , '''10'''  (1976)  pp. 436–496  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''40''' :  3  (1976)  pp. 488–526</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.L. Kantor,  "Certain generalizations of Jordan algebras"  ''Trudy Sem. Vektor. Tenzor. Anal.'' , '''16'''  (1972)  pp. 407–499  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.G. Kac,  "Simple irreducible graded Lie algebras of finite growth"  ''Math. USSR Izv.'' , '''2'''  (1968)  pp. 1271–1312  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''32''' :  6  (1968)  pp. 1323–1367</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.I. Kostrikin,  I.R. Shafarevich,  "Graded Lie algebras of finite characteristic"  ''Math. USSR Izv.'' , '''3'''  (1969)  pp. 237–304  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''33''' :  2  (1969)  pp. 252–322</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
  
 +
====Comments====
 +
The Lie algebras  $  \mathfrak g (A) $
 +
are called Kac–Moody algebras; they have close connections with many areas of mathematics and mathematical physics (cf. [[#References|[a2]]]).
  
 +
There is a second notion which also sometimes goes by the name of graded Lie algebra. This is a  $  \mathbf Z $-
 +
or  $  \mathbf Z / (2) $-
 +
graded vector space  $  V = \oplus V _ {i} $
 +
with a multiplication
  
====Comments====
+
$$
The Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430183.png" /> are called Kac–Moody algebras; they have close connections with many areas of mathematics and mathematical physics (cf. [[#References|[a2]]]).
+
[ , ] :  V \times V  \rightarrow  V
 +
$$
  
There is a second notion which also sometimes goes by the name of graded Lie algebra. This is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430184.png" />- or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430185.png" />-graded vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430186.png" /> with a multiplication
+
such 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/l058430/l058430187.png" /></td> </tr></table>
+
$$
 +
[ V _ {i} , V _ {j} ]  \subset  V _ {i+j} ,
 +
$$
  
such that
+
$$
 +
[ x , y ]  =  ( - 1 )  ^ {ij+1} [ y , x ] ,
 +
$$
  
<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/l058430/l058430188.png" /></td> </tr></table>
+
for all  $  x \in V _ {i} $,
 +
$  y \in V _ {j} $,
 +
and
  
<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/l058430/l058430189.png" /></td> </tr></table>
+
$$
 +
( - 1 )  ^ {ik}
 +
[ [ x , y ] , z ] + ( - 1 )  ^ {ji} [ [ y , z ] , x ] +
 +
( - 1 )  ^ {kj} [ [ z , x ] , y ]  = 0 ,
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430190.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430191.png" />, and
+
for all $  x \in V _ {i} $,
 +
$  y \in V _ {j} $,
 +
$  z \in V _ {k} $.  
 +
One also says that  $  V = \oplus V _ {i} $
 +
has been equipped with a graded Lie product or graded Lie bracket.
  
<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/l058430/l058430192.png" /></td> </tr></table>
+
Graded Lie brackets naturally arise, for instance, in cohomology theory in the context of deformations of algebras and complex structures, [[#References|[a4]]]. A graded vector space  $  V $
 +
with a graded Lie bracket is not a Lie algebra in the usual sense of the word. A Lie superalgebra is a  $  \mathbf Z / (2) $-
 +
graded vector space with a  $  \mathbf Z / (2) $-
 +
graded Lie bracket.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430193.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430194.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430195.png" />. One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430196.png" /> has been equipped with a graded Lie product or graded Lie bracket.
+
Of fundamental importance in recent progress in quantum field theory is the Virasoro algebra. This is a ( $  \mathbf Z $-
 +
graded) Lie algebra with a basis  $  L _ {k} $(
 +
$  k \in \mathbf Z $)
 +
and  $  c $,
 +
and the following commutation relations:
  
Graded Lie brackets naturally arise, for instance, in cohomology theory in the context of deformations of algebras and complex structures, [[#References|[a4]]]. A graded vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430197.png" /> with a graded Lie bracket is not a Lie algebra in the usual sense of the word. A Lie superalgebra is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430198.png" />-graded vector space with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430199.png" />-graded Lie bracket.
+
$$
 +
[ L _ {m} , L _ {n} ] = \
 +
( m - n ) L _ {m+n} +
  
Of fundamental importance in recent progress in quantum field theory is the Virasoro algebra. This is a (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430200.png" />-graded) Lie algebra with a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430201.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430202.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430203.png" />, and the following commutation relations:
+
\frac{m  ^ {3} - m }{12}
  
<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/l058430/l058430204.png" /></td> </tr></table>
+
\delta _ {m , - n }  c .
 +
$$
  
 
See [[#References|[a1]]].
 
See [[#References|[a1]]].

Latest revision as of 16:40, 31 March 2020


A Lie algebra $ \mathfrak g $ over a field $ K $ that is graded by means of an Abelian group $ A $, that is, which splits into a direct sum of subspaces $ \mathfrak g _ \alpha $, $ \alpha \in A $, in such a way that $ [ \mathfrak g _ \alpha , \mathfrak g _ \beta ] \subseteq \mathfrak g _ {\alpha + \beta } $. If $ A $ is an ordered group, then for every filtered Lie algebra (cf. Filtered algebra) the graded algebra associated with it is a graded Lie algebra.

Graded Lie algebras play an important role in the classification of simple finite-dimensional Lie algebras, Jordan algebras and their generalizations, and primitive pseudo-groups of transformations (see [3], [4]). For any semi-simple real Lie algebra its Cartan decomposition can be regarded as a $ \mathbf Z _ {2} $- grading. The local classification of symmetric Riemannian spaces reduces to the classification of $ \mathbf Z _ {2} $- graded simple complex Lie algebras [6].

Some constructions of graded Lie algebras.

1) Let $ U $ be an associative algebra (cf. Associative rings and algebras) endowed with an increasing filtration $ ( U _ {k} : k \in \mathbf Z ) $, suppose that $ [ U _ {k} , U _ {l} ] \subset U _ {k+l-d} $, where $ d $ is a fixed natural number, and let $ \mathfrak u _ {k} = U _ {k+d} / U _ {k+d-1} $. Then the commutation operation in $ U $ induces in the space $ \mathfrak u = \sum _ {k = - \infty } ^ \infty \mathfrak u _ {k} $ the structure of a $ \mathbf Z $- graded Lie algebra. In this way one can obtain some Lie algebras of functions with the Poisson brackets as commutator. In the next two examples, $ U _ {k} = U _ {1} ^ {k} $ for $ k > 0 $ and $ U _ {k} = 0 $ for $ k < 0 $.

a) Let $ U $ be the algebra of linear differential operators with polynomial coefficients and let $ U _ {1} $ be the subspace spanned by its generators $ p _ {i} = \partial / {\partial x _ {i} } $, $ q _ {i} = x _ {i} $, $ i = 1 \dots m $. Then $ [ U _ {k} , U _ {l} ] \subset U _ {k+l-2} $ and $ \mathfrak u $ is the Lie algebra of polynomials in $ p _ {i} $ and $ q _ {i} $ with the usual Poisson brackets.

b) Let $ U $ be the universal enveloping algebra of a finite-dimensional Lie algebra $ \mathfrak g $ and let $ U _ {1} = \mathfrak g $. Then $ [ U _ {k} , U _ {l} ] \subset U _ {k+l-1} $ and $ \mathfrak u $ is canonically isomorphic (as a vector space) to the symmetric algebra over $ \mathfrak g $, that is, to the algebra of polynomials on the dual space $ \mathfrak g ^ {*} $( the Poincaré–Birkhoff–Witt theorem). If $ \mathfrak g $ is the Lie algebra of a connected Lie group $ G $, then the commutator of elements of $ \mathfrak u $ can be interpreted either as the Poisson brackets for the corresponding left-invariant functions on the cotangent bundle $ T ^ {*} G $, or as the Poisson brackets on each orbit of the co-adjoint representation, defined by means of the standard symplectic structure on these orbits.

2) Suppose that $ \mathop{\rm char} k \neq 2 $ and that $ E $ is an $ n $- dimensional vector space over $ k $ endowed with a non-singular quadratic form $ Q $; let $ e _ {1} \dots e _ {n} $ be an orthogonal basis of $ E $. The decomposition of the Clifford algebra $ C (Q) $ into the sum of one-dimensional subspaces $ \langle e _ {i _ {1} } \dots e _ {i _ {k} } \rangle $, $ i _ {1} < \dots < i _ {k} $, is a $ \mathbf Z _ {2} ^ {n} $- grading of it. For $ n = 2m $ the elements of the algebra $ C (Q) $ with zero trace form a simple graded Lie algebra of type $ A _ {N} $, $ N = 2 ^ {m} - 1 $; its grading has a high degree of symmetry; in particular, all graded subspaces are equivalent. Similar gradings (by means of various finite groups) exist for other simple Lie algebras [1].

3) To every Lie pseudo-group of transformations corresponds a Lie algebra of vector fields. The germ $ l $ of this Lie algebra at any point has a natural $ \mathbf Z $- filtration

$$ l = l _ {-1} \supset l _ {0} \supset l _ {1} \supset \dots , $$

where $ l _ {k} $ contains the germs of those vector fields whose coordinates can be expanded in power series without terms of degree less than $ k + 1 $. The associated graded Lie algebra can be interpreted as a Lie algebra of polynomial vector fields.

The classification of simple graded Lie algebras.

To simple primitive Lie pseudo-groups correspond the following four series of simple infinite-dimensional graded Lie algebras (see [5]):

$ W _ {n} $, the Lie algebra of all polynomial vector fields in the $ n $- dimensional affine space;

$ S _ {n} $, its subalgebra consisting of vector fields with zero divergence;

$ H _ {n} $, where $ n = 2 m $, the subalgebra consisting of vector fields that annihilate the differential form

$$ \sum _ { i=1 } ^ { m } d x _ {i} \wedge d x _ {m+i} $$

(Hamiltonian vector fields);

$ K _ {n} $, where $ n = 2 m + 1 $, the subalgebra consisting of vector fields that multiply the differential form

$$ \sum _ { i=1 } ^ { m } ( x _ {m+i} d x _ {i} - x _ {i} d x _ {m+i} ) + d x _ {n} $$

by a function.

Over fields of characteristic $ p > 0 $ one can define simple finite-dimensional graded Lie algebras analogous to $ W _ {n} $, $ S _ {n} $, $ H _ {n} $, and $ K _ {n} $( see [5]).

Simple graded Lie algebras of another type are obtained in the following way [4]. Let $ \mathfrak g = \mathfrak g (A) $ be the Lie algebra defined by means of an indecomposable Cartan matrix $ A = \| a _ {ij} \| $, $ i , j = 1 \dots n $( from now on the notation of the article Cartan matrix is used). The algebra $ \mathfrak g $ is endowed with a $ \mathbf Z ^ {k} $- grading so that $ h _ {i} \in \mathfrak g _ {0} $, $ e _ {i} \in \mathfrak g _ {\alpha _ {i} } $, $ f _ {i} \in \mathfrak g _ {- \alpha _ {i} } $, where $ \alpha _ {i} $ is the row $ ( 0 \dots 1 \dots 0 ) $ with $ 1 $ in the $ i $- th place. Elements $ \alpha \in \mathbf Z ^ {n} $ for which $ \mathfrak g _ \alpha \neq 0 $ are called roots, and the $ \alpha _ {i} $ are called simple roots. Any root is a linear combination of simple roots with integer coefficients of the same sign and $ \mathop{\rm dim} \mathfrak g _ \alpha < \infty $ for any $ \alpha \in \mathbf Z ^ {n} $. The quotient algebra $ {\mathfrak g } ^ \prime (A) $ of $ \mathfrak g $ with respect to its centre, which lies in $ \mathfrak g _ {0} $, is simple as a graded algebra, that is, it does not have non-trivial graded ideals.

Let $ R $ be the totality of linear combinations of rows of the matrix $ A $ with positive coefficients. Then one of the following cases holds:

(P) $ R $ contains a row all elements of which are positive;

(Z) $ R $ contains a zero row;

(N) $ R $ contains a row all elements of which are negative.

In the case (P), $ \mathfrak g (A) = \mathfrak g ^ \prime (A) $ is a simple finite-dimensional Lie algebra. In the case (N), $ \mathfrak g (A) $ is a simple infinite-dimensional Lie algebra. In the case (Z), the algebra $ \mathfrak g ^ \prime = \mathfrak g ^ \prime (A) $ is simple only as a graded algebra. It can be converted in a $ K [ u , u ^ {-1} ] $- algebra so that: a) $ u \mathfrak g _ \alpha ^ \prime = \mathfrak g _ {\alpha + \nu } ^ \prime $, where $ \nu $ is a row of positive numbers; and b) the quotient algebra $ \mathfrak g ^ \prime / ( 1 - u ) \mathfrak g ^ \prime = \overline{ {\mathfrak g }}\; $ is a simple finite-dimensional Lie algebra. The greatest common divisor of all components $ \nu _ {i} $ of the row $ \nu $, which is equal to 1, 2 or 3, is called the index of the algebra $ \mathfrak g ^ \prime $.

The following table is a list of all simple graded Lie algebras with Cartan matrix of type (Z). Here the algebra $ \mathfrak g ^ \prime $ is denoted by the same symbol as the associated simple finite-dimensional Lie algebra $ \overline{ {\mathfrak g }}\; $, but with the addition of its index, given in brackets.

The diagram of simple roots describes the matrix $ A $. Its vertices correspond to the simple roots; the $ i $- th and $ j $- th vertices are joined by an $ ( a _ {ij} a _ {ji} ) $- multiple edge, directed from the $ i $- th vertex to the $ j $- th if $ | a _ {ij} | > | a _ {ji} | $, and undirected if $ | a _ {ij} | = | a _ {ji} | $. Above the vertices stand the numbers $ \nu _ {i} $.

<tbody> </tbody>
Notation Diagram of simple roots
$ A _ {n-1} ^ {(1)} $,

$ n \geq 3 $

$ A _ {1} ^ {(1)} $

$ A _ {2n-2} ^ {(2)} $,

$ n \geq 3 $

$ A _ {2} ^ {(2)} $

$ A _ {2n-3} ^ {(2)} $,

$ n \geq 4 $

$ B _ {n-1} ^ {(1)} $,

$ n \geq 4 $

$ C _ {n-1} ^ {(1)} $,

$ n \geq 3 $

$ D _ {n-1} ^ {(1)} $,

$ n \geq 5 $

$ D _ {n} ^ {(2)} $,

$ n \geq 3 $

$ D _ {4} ^ {(3)} $

$ E _ {6} ^ {(1)} $

$ E _ {6} ^ {(2)} $

$ E _ {7} ^ {(1)} $

$ E _ {8} ^ {(1)} $

$ F _ {4} ^ {(1)} $

$ G _ {2} ^ {(1)} $

By means of graded Lie algebras with Cartan matrix of type (Z) one can classify $ \mathbf Z _ {m} $- graded simple finite-dimensional Lie algebras (see [4], [2]). Namely, let $ \mathfrak g ^ \prime = \mathfrak g ^ \prime (A) $, where $ A $ satisfies condition (Z), and let $ p : \mathbf Z ^ {n} \rightarrow \mathbf Z $ be a homomorphism such that $ p ( \alpha _ {i} ) \geq 0 $ and $ p ( \nu ) = m $. Then $ \mathfrak g _ {k} ^ \prime = \sum _ {p ( \alpha ) = k } \mathfrak g _ \alpha ^ \prime $ for any $ k \in \mathbf Z $ is mapped isomorphically onto the subspace $ \overline{ {\mathfrak g }}\; _ {k} \subset \overline{ {\mathfrak g }}\; $, which depends only on the residue of $ k $ modulo $ m $, and the decomposition $ \overline{ {\mathfrak g }}\; = \sum _ {k=0} ^ {m-1} \overline{ {\mathfrak g }}\; _ {k} $ is a $ \mathbf Z _ {m} $- grading of $ \overline{ {\mathfrak g }}\; $. If the field $ K $ is algebraically closed, then by the method described one obtains, without repetition, all $ \mathbf Z _ {m} $- graded simple finite-dimensional Lie algebras over $ K $. The index of $ \mathfrak g ^ \prime $ is equal to the order of the automorphism $ \theta : x \mapsto ( \mathop{\rm exp} ( {2 \pi i k } / m )) x $, $ x \in \mathfrak g _ {k} $, of the algebra $ \overline{ {\mathfrak g }}\; $ modulo the group of inner automorphisms.

There is a classification of simple $ \mathbf Z $- graded Lie algebras $ \mathfrak g = \sum _ {k = - \infty } ^ \infty \mathfrak g _ {k} $ satisfying the conditions: a) $ \mathop{\rm dim} \mathfrak g _ {k} \leq C | k | ^ {N} $ for some $ C $ and $ N $; b) $ \mathfrak g $ is generated by the subspace $ \mathfrak g _ {-1} + \mathfrak g _ {0} + \mathfrak g _ {1} $; and c) the representation of $ \mathfrak g _ {0} $ on $ \mathfrak g _ {-1} $ is irreducible. In this case either $ \mathfrak g $ is finite-dimensional or it is one of the algebras $ W _ {n} $, $ S _ {n} $, $ H _ {n} $, $ K _ {n} $, or it is the algebra $ \mathfrak g ^ \prime (A) $ defined by a Cartan matrix of type (Z), endowed with a suitable $ \mathbf Z $- grading [4].

A Lie superalgebra is sometimes called a $ \mathbf Z _ {2} $- graded Lie algebra.

References

[1] A.V. Alekseevskii, "Finite commutative Jordan subgroups of complex simple Lie groups" Funct. Anal. Appl. , 8 : 4 (1974) pp. 277–279 Funktsional. Anal. Prilozhen. , 8 : 4 (1974) pp. 1–4
[2] E.B. Vinberg, "The Weyl group of a graded Lie algebra" Math. USSR Izv. , 10 (1976) pp. 436–496 Izv. Akad. Nauk SSSR Ser. Mat. , 40 : 3 (1976) pp. 488–526
[3] I.L. Kantor, "Certain generalizations of Jordan algebras" Trudy Sem. Vektor. Tenzor. Anal. , 16 (1972) pp. 407–499 (In Russian)
[4] V.G. Kac, "Simple irreducible graded Lie algebras of finite growth" Math. USSR Izv. , 2 (1968) pp. 1271–1312 Izv. Akad. Nauk SSSR Ser. Mat. , 32 : 6 (1968) pp. 1323–1367
[5] A.I. Kostrikin, I.R. Shafarevich, "Graded Lie algebras of finite characteristic" Math. USSR Izv. , 3 (1969) pp. 237–304 Izv. Akad. Nauk SSSR Ser. Mat. , 33 : 2 (1969) pp. 252–322
[6] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)

Comments

The Lie algebras $ \mathfrak g (A) $ are called Kac–Moody algebras; they have close connections with many areas of mathematics and mathematical physics (cf. [a2]).

There is a second notion which also sometimes goes by the name of graded Lie algebra. This is a $ \mathbf Z $- or $ \mathbf Z / (2) $- graded vector space $ V = \oplus V _ {i} $ with a multiplication

$$ [ , ] : V \times V \rightarrow V $$

such that

$$ [ V _ {i} , V _ {j} ] \subset V _ {i+j} , $$

$$ [ x , y ] = ( - 1 ) ^ {ij+1} [ y , x ] , $$

for all $ x \in V _ {i} $, $ y \in V _ {j} $, and

$$ ( - 1 ) ^ {ik} [ [ x , y ] , z ] + ( - 1 ) ^ {ji} [ [ y , z ] , x ] + ( - 1 ) ^ {kj} [ [ z , x ] , y ] = 0 , $$

for all $ x \in V _ {i} $, $ y \in V _ {j} $, $ z \in V _ {k} $. One also says that $ V = \oplus V _ {i} $ has been equipped with a graded Lie product or graded Lie bracket.

Graded Lie brackets naturally arise, for instance, in cohomology theory in the context of deformations of algebras and complex structures, [a4]. A graded vector space $ V $ with a graded Lie bracket is not a Lie algebra in the usual sense of the word. A Lie superalgebra is a $ \mathbf Z / (2) $- graded vector space with a $ \mathbf Z / (2) $- graded Lie bracket.

Of fundamental importance in recent progress in quantum field theory is the Virasoro algebra. This is a ( $ \mathbf Z $- graded) Lie algebra with a basis $ L _ {k} $( $ k \in \mathbf Z $) and $ c $, and the following commutation relations:

$$ [ L _ {m} , L _ {n} ] = \ ( m - n ) L _ {m+n} + \frac{m ^ {3} - m }{12} \delta _ {m , - n } c . $$

See [a1].

References

[a1] V.G. Kac, A.K. Raina, "Bombay lectures on highest weight representations" , World Sci. (1987)
[a2] V.G. Kac, "Infinite-dimensional Lie algebras" , Cambridge Univ. Press (1985)
[a3] O. Mathieu, "Classification des algèbres de Lie graduées simples de croissance " Invent. Math. , 86 (1986) pp. 371–426
[a4] M. Hazewinkel (ed.) M. Gerstenhaber (ed.) , Deformation theory of algebras and structures and applications , Kluwer (1988)
How to Cite This Entry:
Lie algebra, graded. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_graded&oldid=17705
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article