Universal enveloping algebra
of a Lie algebra over a commutative ring
with a unit element
The associative -algebra
with a unit element, together with a mapping
for which the following properties hold: 1)
is a homomorphism of Lie algebras, i.e.
is
-linear and
,
; 2) for every associative
-algebra
with a unit element and every
-algebra mapping
such that
,
, there exists a unique homomorphism of associative algebras
, mapping the unit to the unit, such that
. The universal enveloping algebra is unique up to an isomorphism and always exists: If
is the tensor algebra of the
-module
,
is the two-sided ideal generated by all elements of the form
,
, and
is the canonical mapping, then
is the universal enveloping algebra of
.
If is Noetherian and the module
has finite order, then the algebra
is left and right Noetherian. If
is a free module over an integral domain
, then
has no zero divisors. For any finite-dimensional Lie algebra
over a field
the algebra
satisfies the Ore condition (cf. Imbedding of semi-groups) and so has a skew-field of fractions.
If is any
-module, then every homomorphism of Lie algebras
extends to a homomorphism of associative algebras
. This establishes an isomorphism of the category of
-modules and the category of left
-modules, whose existence forms the basis for the application of universal enveloping algebras in the theory of representations of Lie algebras (cf. [3], [4]).
The universal enveloping algebra of the direct product of Lie algebras is the tensor product of the algebras
. If
is a subalgebra of
, where
and
are free
-modules, then the canonical homomorphism
is an imbedding. If
is an extension of the field
, then
. A universal enveloping algebra has a canonical filtration
, where
and
for
is the
-submodule of
generated by the products
,
,
for all
. The graded algebra
associated to this filtration is commutative and is generated by the image under the natural homomorphism
; this mapping defines a homomorphism
of the symmetric algebra
of the
-module
onto
. By the Poincaré–Birkhoff–Witt theorem,
is an algebra isomorphism if
is a free
-module. The following is an equivalent formulation: If
is a basis of the
-module
, where
is a totally ordered set, then the family of monomials
,
,
, forms a basis of the
-module
(in particular,
is injective).
Let be the centre of
. Then for any finite-dimensional Lie algebra
over a field of characteristic zero,
consists of the subalgebra of
-invariant elements in
. If
is semi-simple, then
is the algebra of polynomials in
variables.
One of the important directions of research in universal enveloping algebra is the study of primitive ideals (cf. Primitive ideal; [3]).
References
[1] | N. Bourbaki, "Lie groups and Lie algebras" , Elements of mathematics , Hermann (1975) pp. Chapts. 1–3 (Translated from French) |
[2] | N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII |
[3] | J. Dixmier, "Enveloping algebras" , North-Holland (1977) (Translated from French) |
[4] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) |
[5] | I.M. Gel'fand, "The centre of an infinitesimal group ring" Mat. Sb. , 26 (1950) pp. 103–112 (In Russian) |
[6] | J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) |
Comments
References
[a1] | J.C. Jantzen, "Einhüllende Algebren halbeinfacher Lie-Algebren" , Springer (1983) |
Universal enveloping algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_enveloping_algebra&oldid=41201