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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.==

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 $.

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|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|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  $  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 [[#References|[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 [[#References|[5]]]).

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  $  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} $.
<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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</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/" />

</td> </tr> </tbody> </table>

</td></tr> </table>

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  $  \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  $  \mathbf Z _ {2} $-
graded Lie algebra.

====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>

====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

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

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, [[#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.

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 [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.G. Kac,   A.K. Raina,   "Bombay lectures on highest weight representations" , World Sci.  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.G. Kac,   "Infinite-dimensional Lie algebras" , Cambridge Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Mathieu,   "Classification des algèbres de Lie graduées simples de croissance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058430/l058430205.png" />"  ''Invent. Math.'' , '''86'''  (1986)  pp. 371–426</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)</TD></TR></table>
[[Category:Nonassociative rings and algebras]]