Namespaces
Variants
Actions

Universal enveloping algebra

From Encyclopedia of Mathematics
Jump to: navigation, search

of a Lie algebra over a commutative ring \mathbb{k} with a unit element

The associative \mathbb{k} -algebra U(\mathfrak{g}) with a unit element, together with a mapping \sigma: \mathfrak{g} \to U(\mathfrak{g}) for which the following properties hold:

  1. \sigma is a homomorphism of Lie algebras, i.e., \sigma is a \mathbb{k} -linear transformation and \sigma([x,y]) = \sigma(x) \sigma(y) - \sigma(y) \sigma(x) for all x,y \in \mathfrak{g} .
  2. For every associative \mathbb{k} -algebra A with a unit element and every \mathbb{k} -algebra homomorphism \alpha: \mathfrak{g} \to A such that \alpha([x,y]) = \alpha(x) \alpha(y) - \alpha(y) \alpha(x) for all x,y \in \mathfrak{g} , there exists a unique homomorphism of associative algebras \beta: U(\mathfrak{g}) \to A , mapping the unit to the unit, such that \alpha = \beta \circ \sigma .

The universal enveloping algebra is unique up to an isomorphism and always exists: If T(\mathfrak{g}) is the tensor algebra of the \mathbb{k} -module \mathfrak{g} , I is the two-sided ideal generated by all elements of the form [x,y] - (x \otimes y - y \otimes x) for x,y \in \mathfrak{g} , and \sigma: \mathfrak{g} \to T(\mathfrak{g}) / I is the canonical map, then T(\mathfrak{g}) / I is the universal enveloping algebra of \mathfrak{g} .

If \mathbb{k} is Noetherian and the module \mathfrak{g} has finite order, then the algebra U(\mathfrak{g}) is left- and right-Noetherian. If \mathfrak{g} is a free module over an integral domain \mathbb{k} , then U(\mathfrak{g}) has no zero divisors. For any finite-dimensional Lie algebra \mathfrak{g} over a field \mathbb{k} , the algebra U(\mathfrak{g}) satisfies the Ore condition (cf. imbedding of semi-groups) and so has a skew-field of fractions.

If V is any \mathbb{k} -module, then every Lie-algebra homomorphism \mathfrak{g} \to \operatorname{End} V extends to a homomorphism of associative algebras U(\mathfrak{g}) \to \operatorname{End} V . This establishes an isomorphism between the category of \mathfrak{g} -modules and the category of left U(\mathfrak{g}) -modules, whose existence forms the basis for the application of universal enveloping algebras in the theory of representations of Lie algebras ([3], [4]).

The universal enveloping algebra of the direct product of Lie algebras \mathfrak{g}_{1},\ldots,\mathfrak{g}_{n} is the tensor product of the U(\mathfrak{g}_{i}) ’s. If \mathfrak{h} is a subalgebra of \mathfrak{g} , where \mathfrak{h} and \mathfrak{g} / \mathfrak{h} are free \mathbb{k} -modules, then the canonical homomorphism U(\mathfrak{h}) \to U(\mathfrak{g}) is an imbedding. If \mathbb{k}' is an extension of the field \mathbb{k} , then U(\mathfrak{g} \otimes_{\mathbb{k}} \mathbb{k}') = U(\mathfrak{g}) \otimes_{\mathbb{k}} \mathbb{k}' . A universal enveloping algebra has a canonical filtration {U_{0}}(\mathfrak{g}) \subseteq {U_{1}}(\mathfrak{g}) \subseteq \ldots , where {U_{0}}(\mathfrak{g}) = \mathbb{k} \cdot 1 and {U_{n}}(\mathfrak{g}) for n \in \mathbf{N} is the \mathbb{k} -submodule of U(\mathfrak{g}) generated by the products \sigma(x_{1}) \cdots \sigma(x_{m}) , where m \in \mathbf{N}_{\leq n} and x_{i} \in \mathfrak{g} for all i \in \mathbf{N}_{\leq n} . The graded algebra \operatorname{gr} U(\mathfrak{g}) associated to this filtration is commutative and is generated by the image under the natural homomorphism \mathfrak{g} \to \operatorname{gr} U(\mathfrak{g}) ; this mapping defines a homomorphism \delta of the symmetric algebra S(\mathfrak{g}) of the \mathbb{k} -module \mathfrak{g} onto \operatorname{gr} U(\mathfrak{g}) . By the Poincaré-Birkhoff-Witt theorem, \delta: S(\mathfrak{g}) \to \operatorname{gr} U(\mathfrak{g}) is an algebra isomorphism if \mathfrak{g} is a free \mathbb{k} -module. The following is an equivalent formulation: If (x_{i})_{i \in I} is an ordered basis of the \mathbb{k} -module \mathfrak{g} , where I is a totally ordered set, then the family of monomials \sigma(x_{i_{1}}) \cdots \sigma(x_{i_{n}}) , for i_{1} \leq_{I} \ldots \leq_{I} i_{n} and n \in \mathbf{N}_{0} , forms a basis of the \mathbb{k} -module U(\mathfrak{g}) (in particular, \sigma is injective).

Let Z(\mathfrak{g}) be the centre of U(\mathfrak{g}) . Then for any finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero, \operatorname{gr} Z(\mathfrak{g}) \subseteq \operatorname{gr} U(\mathfrak{g}) = S(\mathfrak{g}) consists of the subalgebra of G -invariant elements of S(\mathfrak{g}) . If \mathfrak{g} is semi-simple, then Z(\mathfrak{g}) is the algebra of polynomials in \operatorname{rank}(\mathfrak{g}) variables.

One of the important directions of research in universal enveloping algebras is the study of primitive ideals ([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)
[a1] J.C. Jantzen, “Einhüllende Algebren halbeinfacher Lie-Algebren”, Springer (1983).
How to Cite This Entry:
Universal enveloping algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_enveloping_algebra&oldid=53586
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article