Universal enveloping algebra

of a Lie algebra $ \mathfrak{g} $ 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).