Lie algebra, graded

A Lie algebra over a field that is graded by means of an Abelian group , that is, which splits into a direct sum of subspaces , , in such a way that . If 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 -grading. The local classification of symmetric Riemannian spaces reduces to the classification of -graded simple complex Lie algebras [6].

Some constructions of graded Lie algebras.

1) Let be an associative algebra (cf. Associative rings and algebras) endowed with an increasing filtration , suppose that , where is a fixed natural number, and let . Then the commutation operation in induces in the space the structure of a -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, for and for .

a) Let be the algebra of linear differential operators with polynomial coefficients and let be the subspace spanned by its generators , , . Then and is the Lie algebra of polynomials in and with the usual Poisson brackets.

b) Let be the universal enveloping algebra of a finite-dimensional Lie algebra and let . Then and is canonically isomorphic (as a vector space) to the symmetric algebra over , that is, to the algebra of polynomials on the dual space (the Poincaré–Birkhoff–Witt theorem). If is the Lie algebra of a connected Lie group , then the commutator of elements of can be interpreted either as the Poisson brackets for the corresponding left-invariant functions on the cotangent bundle , 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 and that is an -dimensional vector space over endowed with a non-singular quadratic form ; let be an orthogonal basis of . The decomposition of the Clifford algebra into the sum of one-dimensional subspaces , , is a -grading of it. For the elements of the algebra with zero trace form a simple graded Lie algebra of type , ; 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 of this Lie algebra at any point has a natural -filtration

where contains the germs of those vector fields whose coordinates can be expanded in power series without terms of degree less than . 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]):

, the Lie algebra of all polynomial vector fields in the -dimensional affine space;

, its subalgebra consisting of vector fields with zero divergence;

, where , the subalgebra consisting of vector fields that annihilate the differential form

(Hamiltonian vector fields);

, where , the subalgebra consisting of vector fields that multiply the differential form

by a function.

Over fields of characteristic one can define simple finite-dimensional graded Lie algebras analogous to , , , and (see [5]).

Simple graded Lie algebras of another type are obtained in the following way [4]. Let be the Lie algebra defined by means of an indecomposable Cartan matrix , (from now on the notation of the article Cartan matrix is used). The algebra is endowed with a -grading so that , , , where is the row with in the -th place. Elements for which are called roots, and the are called simple roots. Any root is a linear combination of simple roots with integer coefficients of the same sign and for any . The quotient algebra of with respect to its centre, which lies in , is simple as a graded algebra, that is, it does not have non-trivial graded ideals.

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

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

(Z) contains a zero row;

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

In the case (P), is a simple finite-dimensional Lie algebra. In the case (N), is a simple infinite-dimensional Lie algebra. In the case (Z), the algebra is simple only as a graded algebra. It can be converted in a -algebra so that: a) , where is a row of positive numbers; and b) the quotient algebra is a simple finite-dimensional Lie algebra. The greatest common divisor of all components of the row , which is equal to 1, 2 or 3, is called the index of the algebra .

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

The diagram of simple roots describes the matrix . Its vertices correspond to the simple roots; the -th and -th vertices are joined by an -multiple edge, directed from the -th vertex to the -th if , and undirected if . Above the vertices stand the numbers .'

<tbody> </tbody> Notation Diagram of simple roots , , , , , , ,

By means of graded Lie algebras with Cartan matrix of type (Z) one can classify -graded simple finite-dimensional Lie algebras (see [4], [2]). Namely, let , where satisfies condition (Z), and let be a homomorphism such that and . Then for any is mapped isomorphically onto the subspace , which depends only on the residue of modulo , and the decomposition is a -grading of . If the field is algebraically closed, then by the method described one obtains, without repetition, all -graded simple finite-dimensional Lie algebras over . The index of is equal to the order of the automorphism , , of the algebra modulo the group of inner automorphisms.

There is a classification of simple -graded Lie algebras satisfying the conditions: a) for some and ; b) is generated by the subspace ; and c) the representation of on is irreducible. In this case either is finite-dimensional or it is one of the algebras , , , , or it is the algebra defined by a Cartan matrix of type (Z), endowed with a suitable -grading [4].

A Lie superalgebra is sometimes called a -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 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 - or -graded vector space with a multiplication

such that

for all , , and

for all , , . One also says that 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 with a graded Lie bracket is not a Lie algebra in the usual sense of the word. A Lie superalgebra is a -graded vector space with a -graded Lie bracket.

Of fundamental importance in recent progress in quantum field theory is the Virasoro algebra. This is a (-graded) Lie algebra with a basis () and , and the following commutation relations:

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)