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 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)