nLab
universal enveloping algebra

Definition

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.

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

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 $\mathrm{GK}(U(L))$.

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 $C$.

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

An oidification is the universal enveloping algebroid.

Literature

• Related entries: Hausdorff series
• 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
• MathOverflow question What is the universal enveloping algebra which is looking for a rather general construction in a class of symmetric monoidal pseudoabelian categories.