# 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)

#### References

 [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=41201
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article