# Universal enveloping algebra

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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 (, ).

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 ().

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