Lie algebra, graded
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 finitedimensional Lie algebras, Jordan algebras and their generalizations, and primitive pseudogroups of transformations (see [3], [4]). For any semisimple 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].
Contents
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+ld} $, where $ d $ is a fixed natural number, and let $ \mathfrak u _ {k} = U _ {k+d} / U _ {k+d1} $. 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+l2} $ 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 finitedimensional Lie algebra $ \mathfrak g $ and let $ U _ {1} = \mathfrak g $. Then $ [ U _ {k} , U _ {l} ] \subset U _ {k+l1} $ 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 leftinvariant functions on the cotangent bundle $ T ^ {*} G $, or as the Poisson brackets on each orbit of the coadjoint 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 nonsingular 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 onedimensional 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 pseudogroup 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 pseudogroups correspond the following four series of simple infinitedimensional 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 finitedimensional 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 nontrivial 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 finitedimensional Lie algebra. In the case (N), $ \mathfrak g (A) $ is a simple infinitedimensional 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 finitedimensional 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 finitedimensional 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} $.
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By means of graded Lie algebras with Cartan matrix of type (Z) one can classify $ \mathbf Z _ {m} $ graded simple finitedimensional 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} ^ {m1} \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 finitedimensional 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 finitedimensional 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, "Infinitedimensional Lie algebras" , Cambridge Univ. Press (1985) 
[a3]  O. Mathieu, "Classification des algèbres de Lie graduées simples de croissance $\leq 1$" Invent. Math. , 86 (1986) pp. 371–426 
[a4]  M. Hazewinkel (ed.) M. Gerstenhaber (ed.) , Deformation theory of algebras and structures and applications , Kluwer (1988) 
Lie algebra, graded. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_graded&oldid=53486