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].
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 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>
|
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 ![]() |
[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