Contents

Definition

For Lie algebras

Given a Lie algebra $L$ internal to some symmetric monoidal $k$-linear category $C=(C,\otimes ,1,\tau )$, an enveloping monoid (or enveloping algebra) of $L$ in $C$ is any morphism $f:L\to \mathrm{Lie}(A)$ of Lie algebras in $C$ where $A$ is a monoid (= algebra) in $C$, and $\mathrm{Lie}(A)$ is the underlying object of $A$ equipped with the Lie bracket $[,{]}_{\mathrm{Lie}(A)}=\mu -\mu \circ {\tau }_{A,A}$. In further we will just write $A$ for $\mathrm{Lie}(A)$. A morphism of enveloping algebras $\varphi :(f:L\to A)\to (f\prime :L\to A\prime )$ is a morphism $g:A\to A\prime$ of monoids completing a commutative triangle of morphisms in $C$, i.e. $g\circ f=f\prime$. With an obvious composition of morphisms, the enveloping algebras of $L$ form a category. A universal enveloping algebra of $L$ in $C$ is any universal initial object $i_L:L\to U(L)$ in the category of enveloping algebras of $L$; it is of course unique up to an isomorphism if it exists. If it exists for all Lie algebras in $C$, then the rule $L\mapsto U(L)$ can be extended to a functor $U$ which is the left adjoint to the functor $\mathrm{Lie}:A\mapsto \mathrm{Lie}(A)$ defined above and the morphism $i_L:L\to U(L)$ is the unit of the adjunction.

For $L_{\mathrm{\infty }}$-algebras

In the more general context of higher algebra there is a notion of universal enveloping E-n algebra of an L-infinity algebra for all $n\in \mathbb{N}$ which generalizes the notion of universal associative algebra envelope of a Lie algebra. See at universal enveloping E-n algebra.

Existence

The existence of the universal enveloping algebra is easy in many concrete symmetric monoidal categories, e.g. the symmetric monoidal category of bounded chain complexes (giving the universal enveloping dg-algebra of a dg-Lie algebra), but not true in general.

First of all if $C$ admits countable coproducts, form the tensor algebra $\mathrm{TL}={\coprod }_{n=0}^{\mathrm{\infty }}{L}^{\otimes n}$ on the object $L$; this is a monoid in $C$. In most standard cases, one can also form the smallest 2-sided ideal (i.e. $A$-subbimodule) $I$ in monoid $A$ among those ideals whose inclusion into $A$ is factorizing the map $([,]-m_{\mathrm{TL}}+m_{\mathrm{TL}}\circ \tau )\circ \otimes :L\otimes L\to \mathrm{TL}$; if the coequalizers exist in $C$ then we can form the quotient object $\mathrm{TL}/I$ and there is an induced monoid structure in it. Under mild conditions on $C$, the natural morphism $i_L:L\to \mathrm{TL}/I$ is an universal enveloping monoid of $L$ in $C$. If $C$ is an abelian tensor category and some flatness conditions on the tensor product are satisfied, then the enveloping monoid $i_L:L\to \mathrm{TL}/I$ is a monic morphism in $C$ and $U(L\coprod L)\cong U(L)\otimes U(L)$.

Properties

Isomorphism problem

The isomorphism problem for enveloping algebras is about the fact that the universal enveloping monoids of two Lie algebras of $C$ are isomorphic as associative monoids in $C$, but this does not imply that the Lie algebras are isomorphic. This is even not true in general for the Lie $k$-algebras (in classical sense), even if $k$ is a field of characteristics zero. It is known however in that case that the dimension of the finite-dimensional Lie $k$-algebra $L$ can be read off from its universal enveloping $k$-algebra as its Gel’fand-Kirillov dimension $\mathrm{GK}(U(L))$.

Poisson algebra structure on $U(𝔤)$

The universal enveloping algebra $U(𝔤)$ of a Lie algebra is naturally a (non-commutative) Poisson algebra with the restriction of the Poisson bracket to generators being the original Lie bracket

Hopf algebra structure on $U(𝔤)$

Suppose the universal enveloping algebras of Lie algebras exist in a $k$-linear symmetric monoidal category $C$ and the functorial choice $L\mapsto U(L)$ realizing the above construction with tensor products is fixed. For example, this is true in the category of $k$-modules where $k$ is a commutative ring. Then the projection $L\to 0$ where $0$ is the trivial Lie algebra induces the counit $ϵ:U(L)\to U(0)=1$. The coproduct $\Delta :U(L)\to U(L\coprod L)\cong U(L)\otimes U(L)$ is induced by the diagonal map $L\to L\coprod L$ whereas the antipode $S=U(-\mathrm{id}):U(L)\to U(L)$. One checks that these morphisms make $U(L)$ into a Hopf algebra in $C$.

If the category is simply the vector spaces over a field $k$, then for $l\in L$, after we identify $L$ with its image in $U(L)$, $\Delta (l)=l\otimes 1+1\otimes l$, i.e. the elements in $L$ are the primitive elements in $U(L)$.

PBW theorem

The Poincaré–Birkhoff–Witt theorem states that the associated graded algebra of an enveloping algebra $U(g)$ in characteristics zero is canonically isomorphic to a symmetric algebra $\mathrm{Sym}(g)$, and $U(g)$ is isomorphic to $S(g)$ as a coalgebra, via the projection map $U(g)\to \mathrm{Gr}U(g)$.

Examples

Universal enveloping of a tangent Lie algebra

The universal enveloping algebra of the tangent Lie algebra of a finite-dimensional Lie group $G$ over real or complex numbers is canonically isomorphic to the algebra of the left invariant differential operators on $G$.

Related concepts

• Hausdorff series, Poincaré–Birkhoff–Witt theorem, coexponential map, Duflo isomorphism

• An oidification is the universal enveloping algebroid.

• universal enveloping E-n algebra

Literature

• N. Bourbaki, Lie groups and Lie algebras
• Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics 11, Amer. Math. Soc.
• wikipedia: universal enveloping algebra, PBW theorem
• F. A. Berezin, Some remarks about the associated envelope of a Lie algebra, Func. Analysis and Its Appl. 1967, 1:2, 91–102; Rus. original in Fun. Anal. Pril. 1:2 (1967) 1–14 Rus. pdf MR219671
• MathOverflow question What is the universal enveloping algebra which is looking for a rather general construction in a class of symmetric monoidal pseudoabelian categories.