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 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 grading. The local classification of symmetric Riemannian spaces reduces to the classification of graded simple complex Lie algebras [6].
Contents
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 finitedimensional 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 leftinvariant functions on the cotangent bundle , 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 and that is an dimensional vector space over endowed with a nonsingular quadratic form ; let be an orthogonal basis of . The decomposition of the Clifford algebra into the sum of onedimensional 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 pseudogroup 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 pseudogroups correspond the following four series of simple infinitedimensional 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 finitedimensional 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 nontrivial 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 finitedimensional Lie algebra. In the case (N), is a simple infinitedimensional 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 finitedimensional 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 finitedimensional 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 .'
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By means of graded Lie algebras with Cartan matrix of type (Z) one can classify graded simple finitedimensional 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 finitedimensional 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 finitedimensional 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, "Infinitedimensional 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) 
Lie algebra, graded. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_graded&oldid=17705