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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)

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)
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