# Universal enveloping algebra
The **universal enveloping algebra** of a [Lie algebra](https://www.edgepedia.org/lie-algebra) \( \mathfrak{g} \) over a commutative ring \( \mathbb{k} \) with unit is the associative unital \( \mathbb{k} \)-algebra \( U(\mathfrak{g}) \) together with a Lie algebra homomorphism \( \sigma : \mathfrak{g} \to U(\mathfrak{g}) \) that is universal: every Lie algebra homomorphism from \( \mathfrak{g} \) into the underlying Lie algebra of an associative unital algebra \( A \) factors uniquely through \( \sigma \) as an associative algebra homomorphism \( U(\mathfrak{g}) \to A \).[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) Concretely, \( U(\mathfrak{g}) \) is the quotient of the tensor algebra \( T(\mathfrak{g}) \) by the two-sided ideal generated by the elements \( x \otimes y - y \otimes x - [x,y] \) for \( x, y \in \mathfrak{g} \), so that in \( U(\mathfrak{g}) \) the [commutator](https://www.edgepedia.org/commutator) of the images of \( x \) and \( y \) reproduces the Lie bracket.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) • [2](https://mathworld.wolfram.com/UniversalEnvelopingAlgebra.html)
| Key fact | Detail |
|---|---|
| Construction | \( U(\mathfrak{g}) = T(\mathfrak{g})/I \), where \( I \) is generated by \( x \otimes y - y \otimes x - [x,y] \); the object is unique up to isomorphism and always exists.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) |
| Universal property | Any Lie homomorphism \( \mathfrak{g} \to A_L \) (\( A \) associative with unit) extends uniquely to an algebra homomorphism \( U(\mathfrak{g}) \to A \) mapping unit to unit.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) |
| PBW basis | For an ordered basis \( (x_i)_{i \in I} \) of \( \mathfrak{g} \), the ordered monomials \( \sigma(x_{i_1}) \cdots \sigma(x_{i_n}) \) with \( i_1 \leq \cdots \leq i_n \) form a basis; over a field, in particular, \( \sigma \) is injective.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) |
| Associated graded | \( \operatorname{gr} U(\mathfrak{g}) \cong S(\mathfrak{g}) \), the symmetric algebra, which is commutative.[3](https://www.maths.tcd.ie/~mozgovoy/data/notes_LieAlg_2015.pdf) |
| Hopf structure | \( U(\mathfrak{g}) \) is a Hopf algebra with \( \Delta(x) = x \otimes 1 + 1 \otimes x \), \( \varepsilon(x) = 0 \), and antipode \( S(x) = -x \) on \( \mathfrak{g} \).[4](https://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/02SS/QGandNCG.pdf) |
| Ring-theoretic properties | If \( \mathbb{k} \) is Noetherian and \( \mathfrak{g} \) is finite over it, \( U(\mathfrak{g}) \) is left and right Noetherian; if \( \mathfrak{g} \) is free over an integral domain, \( U(\mathfrak{g}) \) has no zero divisors; a finite-dimensional \( U(\mathfrak{g}) \) over a field satisfies the Ore condition and has a skew field of fractions.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) |
| Center (semisimple case) | For semisimple \( \mathfrak{g} \) over a field of characteristic zero, \( Z(\mathfrak{g}) \) is a polynomial algebra in \( \operatorname{rank}(\mathfrak{g}) \) variables.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) |
## Definition and universal property
The defining property has two parts. First, the map \( \sigma \) is \( \mathbb{k} \)-linear and satisfies \( \sigma([x,y]) = \sigma(x)\sigma(y) - \sigma(y)\sigma(x) \), so it is a homomorphism of Lie algebras when the associative algebra \( U(\mathfrak{g}) \) is given the commutator bracket. Second, for every unital associative \( \mathbb{k} \)-algebra \( A \) and every \( \mathbb{k} \)-linear map \( \alpha : \mathfrak{g} \to A \) satisfying the same commutator identity, there exists a unique unital algebra homomorphism \( \beta : U(\mathfrak{g}) \to A \) with \( \alpha = \beta \circ \sigma \).[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) This uniqueness among objects with the property gives the name "universal": \( U(\mathfrak{g}) \) is unique up to isomorphism and always exists.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra)
The construction realizes the universal object directly. The tensor algebra \( T(\mathfrak{g}) = \mathbb{k} \oplus \mathfrak{g} \oplus \mathfrak{g} \otimes \mathfrak{g} \oplus \cdots \) is the free associative algebra on the vector space \( \mathfrak{g} \), and a linear map out of \( \mathfrak{g} \) extends uniquely to an algebra map out of \( T(\mathfrak{g}) \). Quotienting by the ideal generated by \( x \otimes y - y \otimes x - [x,y] \) forces exactly the relations that make the extension respect Lie brackets, and the universal property of the quotient then gives the enveloping algebra.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) • [4](https://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/02SS/QGandNCG.pdf) Equivalently, \( U(\mathfrak{g}) \) is generated as an algebra by the image of \( \mathfrak{g} \).[4](https://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/02SS/QGandNCG.pdf)
## The Poincaré–Birkhoff–Witt theorem
The central structural result is the Poincaré–Birkhoff–Witt theorem. In one form, it states that if \( (x_1, \ldots, x_n) \) is a basis of \( \mathfrak{g} \), then the monomials \( i(x_1)^{k_1} \cdots i(x_n)^{k_n} \) with \( k_1, \ldots, k_n \in \mathbb{N} \) form a basis of \( U(\mathfrak{g}) \), where \( i \) is the canonical map \( \mathfrak{g} \to T(\mathfrak{g}) \to U(\mathfrak{g}) \).[3](https://www.maths.tcd.ie/~mozgovoy/data/notes_LieAlg_2015.pdf) The same statement appears with a totally ordered basis \( (x_i)_{i \in I} \) over an arbitrary index set.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) The theorem has two important consequences. It shows that \( \sigma \) is injective, so \( \mathfrak{g} \) embeds in \( U(\mathfrak{g}) \), and it gives a normal form: every product of basis monomials can be rewritten as a linear combination of ordered monomials using the defining relations \( x_j x_i - x_i x_j = [x_j, x_i] \) for \( j > i \).[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) • [5](https://docs.magma-maths.org/LieTheory/LieAlgebras/universal_enveloping_algebras.html)
The theorem is often stated through a filtration. Defining \( U_m \) as the image of tensors of degree at most \( m \) gives a filtration \( 0 = U_{-1} \subset U_0 \subset U_1 \subset \cdots \) with \( U_m U_n \subset U_{m+n} \). Since \( xy - yx \in U_{m+n-1} \) for \( x \in U_m \), \( y \in U_n \), the associated graded algebra \( \operatorname{gr} U(\mathfrak{g}) \) is commutative, and PBW asserts that the natural map \( S(\mathfrak{g}) \to \operatorname{gr} U(\mathfrak{g}) \) is an isomorphism.[3](https://www.maths.tcd.ie/~mozgovoy/data/notes_LieAlg_2015.pdf) • [6](https://www.math.canterbury.ac.nz/~j.booher/expos/pbw.pdf)
## Representations
The main application of \( U(\mathfrak{g}) \) is to representation theory. If \( V \) is a \( \mathbb{k} \)-module, every Lie algebra homomorphism \( \mathfrak{g} \to \operatorname{End} V \) extends uniquely to an associative algebra homomorphism \( U(\mathfrak{g}) \to \operatorname{End} V \), which establishes an isomorphism between the category of \( \mathfrak{g} \)-modules and the category of left \( U(\mathfrak{g}) \)-modules.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) In particular, \( U(\mathfrak{g}) \) is central to constructing irreducible modules over a semisimple Lie algebra.[7](https://epubs.siam.org/doi/10.1137/1.9781611971330.ch3)
## Hopf algebra structure
\( U(\mathfrak{g}) \) carries a natural [Hopf algebra](https://www.edgepedia.org/hopf-algebra) structure. The diagonal map \( \mathfrak{g} \to \mathfrak{g} \times \mathfrak{g} \) induces a comultiplication with \( \Delta(x) = x \otimes 1 + 1 \otimes x \), the counit satisfies \( \varepsilon(x) = 0 \), and the antipode is given on \( \mathfrak{g} \) by \( S(x) = -x \); coassociativity and the counit and antipode axioms follow from the universal property.[4](https://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/02SS/QGandNCG.pdf) Elements satisfying \( \Delta(x) = x \otimes 1 + 1 \otimes x \) are called primitive, and the primitive elements of a Hopf algebra form a Lie subalgebra under the commutator, so \( \mathfrak{g} \) sits inside \( U(\mathfrak{g}) \) as primitive elements.[4](https://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/02SS/QGandNCG.pdf) The Hopf structure also gives \( U(\mathfrak{g} \times \mathfrak{h}) \cong U(\mathfrak{g}) \otimes U(\mathfrak{h}) \), and the Encyclopedia of Mathematics records the same tensor product rule for direct products of several Lie algebras.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra)
## Triangular decomposition and the center
For a semisimple Lie algebra \( L \) with [Cartan subalgebra](https://www.edgepedia.org/cartan-subalgebra) \( H \) and root space decomposition \( L = L^- \oplus H \oplus L^+ \), PBW-type corollaries give a vector space isomorphism \( U(L) \cong U(L^-) \otimes U(H) \otimes U(L^+) \). Here \( U(H) = S(H) \) is a polynomial algebra \( \mathbb{k}[h_1, \ldots, h_r] \) in the simple coroots, and \( U(L^+) \), \( U(L^-) \) are generated by the simple raising and lowering elements respectively.[3](https://www.maths.tcd.ie/~mozgovoy/data/notes_LieAlg_2015.pdf) • [7](https://epubs.siam.org/doi/10.1137/1.9781611971330.ch3) This decomposition underlies the standard construction of highest-weight modules.
The center is described by the [Harish-Chandra isomorphism](https://www.edgepedia.org/harish-chandra-isomorphism). For finite-dimensional \( \mathfrak{g} \) over a field of characteristic zero, the associated graded of the center \( Z(\mathfrak{g}) \) sits inside \( \operatorname{gr} U(\mathfrak{g}) = S(\mathfrak{g}) \) as the subalgebra of invariants under the adjoint action; when \( \mathfrak{g} \) is semisimple, \( Z(\mathfrak{g}) \) is a polynomial algebra in \( \operatorname{rank}(\mathfrak{g}) \) variables.[1](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra) [Harish-Chandra](https://www.edgepedia.org/harish-chandra) introduced, for any semisimple Lie algebra, an isomorphism from the center of \( U(\mathfrak{g}) \) to the algebra of Weyl-group-invariant elements in the symmetric algebra of a Cartan subalgebra, now known as the Harish-Chandra isomorphism.[8](https://publications.ias.edu/sites/default/files/harish2.pdf)
## Positive characteristic and variants
In characteristic \( p > 0 \), the enveloping algebra is too large for many purposes, although it is not true in general that \( x^p \) is central in it for every \( x \in \mathfrak{g} \).[12](https://www.numdam.org/articles/10.24033/asens.1225/) A restricted Lie algebra comes with a \( p \)-power map \( x \mapsto x^{[p]} \) compatible with the bracket, and the restricted universal enveloping algebra \( u(\mathfrak{g}) \) is the quotient of \( U(\mathfrak{g}) \) by the ideal generated by all \( x^p - x^{[p]} \). If \( \mathfrak{g} \) is \( n \)-dimensional over \( \mathbb{k} \), then \( u(\mathfrak{g}) \) is \( n^p \)-dimensional as a vector space, and the category of restricted \( \mathfrak{g} \)-modules is equivalent to the category of \( u(\mathfrak{g}) \)-modules.[9](https://math.mit.edu/~hrm/palestine/weibel/07-lie_algebra_homology_and_cohomology.pdf) Restricted enveloping algebras also appear in the classification of small Hopf algebras: a Hopf algebra of prime dimension \( p \) over an algebraically closed field of characteristic \( p \) is either a group algebra or a restricted universal enveloping algebra.[10](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/blms.12242?af=R)
The embedding of \( \mathfrak{g} \) in \( U(\mathfrak{g}) \) guaranteed by PBW holds over fields, but can fail over more general base rings. P. M. Cohn constructed an example over a commutative ring in characteristic \( p \) in which a Lie algebra does not embed in its universal enveloping algebra, while the embedding always holds in characteristic zero, where \( U(\mathfrak{g}) \) can be defined as a deformation of \( S(\mathfrak{g}) \).[11](https://mathoverflow.net/questions/25020/what-is-the-universal-enveloping-algebra)
For computation, systems such as [Magma](https://www.edgepedia.org/magma-computer-algebra-system) implement \( U(L) \) with PBW bases and, for semisimple \( L \) in characteristic zero, an integral basis in divided powers \( y_i^{(n)} = y_i^n/n! \) and binomial coefficients \( \binom{h_i}{k} \) built from a Chevalley basis; in this basis the structure constants are integers and multiplication is generally faster than with ordinary PBW bases.[5](https://docs.magma-maths.org/LieTheory/LieAlgebras/universal_enveloping_algebras.html)
## References
1. [Universal enveloping algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Universal_enveloping_algebra)
2. [Universal Enveloping Algebra, Wolfram MathWorld](https://mathworld.wolfram.com/UniversalEnvelopingAlgebra.html)
3. [S. Mozgovoy, Introduction to Lie Algebras, lecture notes (Trinity College Dublin)](https://www.maths.tcd.ie/~mozgovoy/data/notes_LieAlg_2015.pdf)
4. [B. Pareigis, Lectures on Quantum Groups and Noncommutative Geometry, LMU Munich](https://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/02SS/QGandNCG.pdf)
5. [Universal Enveloping Algebras, Magma Handbook](https://docs.magma-maths.org/LieTheory/LieAlgebras/universal_enveloping_algebras.html)
6. [J. Booher, PBW Theorem, University of Canterbury exposition](https://www.math.canterbury.ac.nz/~j.booher/expos/pbw.pdf)
7. [A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods, Ch. 3, SIAM](https://epubs.siam.org/doi/10.1137/1.9781611971330.ch3)
8. [Harish-Chandra: a biographical sketch, Institute for Advanced Study](https://publications.ias.edu/sites/default/files/harish2.pdf)
9. [C. Weibel, An Introduction to Homological Algebra, Ch. 7, Lie algebra homology and cohomology](https://math.mit.edu/~hrm/palestine/weibel/07-lie_algebra_homology_and_cohomology.pdf)
10. [S.-H. Ng, X. Wang, Hopf algebras of prime dimension in positive characteristic, Bulletin of the London Mathematical Society 51 (2019)](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/blms.12242?af=R)
11. [What is the universal enveloping algebra?, MathOverflow](https://mathoverflow.net/questions/25020/what-is-the-universal-enveloping-algebra)
12. [The center of the universal enveloping algebra of a Lie algebra in characteristic $p$](https://www.numdam.org/articles/10.24033/asens.1225/)
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