nLab universal enveloping algebra

Definition

For Lie algebras

Given a Lie algebra object $L$ interna to some symmetric monoidal $k$-linear category $C = (C,\otimes, \mathbf{1},\tau)$, an enveloping monoid (or enveloping algebra) of $L$ in $C$ is any morphism $f \colon L\to Lie(A)$ of Lie algebras in $C$ where $A$ is a monoid (= algebra) in $C$, and $Lie(A)$ is the underlying object of $A$ equipped with the Lie bracket $[,]_{Lie(A)}=\mu-\mu\circ\tau_{A,A}$.

A morphism of enveloping algebras

$$\phi \;\colon\; \big(A, f \colon L\to Lie(A)\big) \longrightarrow \big(A, f' \colon L\to A'\big)$$

is a morphism $g \colon A\to A'$ of monoid objects completing a commutative triangle of morphisms in $C$, i.e. $g\circ f = f'$. 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 $i_L \colon L\to U(L)$ in the category of enveloping algebras of $L$, which implies that it is 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 forgetful functor $Lie \colon A\mapsto Lie(A)$ defined above and the morphism $i_L:L\to U(L)$ is the unit of the adjunction.

Remark

(for super Lie algebras as eg. in algebraic topology)
The internal generality of the above definitions is used notably in algebraic topology where the relevant Lie algebras arise from Whitehead brackets which are super Lie algebras (see there), i.e. Lie algebra objects internal to super vector spaces (cf. eg. Milnor & Moore 1965, §5; May & Ponto 2012, Def. 22.1.3).

For $L_\infty$-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 $TL=\coprod_{n=0}^\infty L^{\otimes n}$ 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 $([,]-m_{TL}+m_{TL}\circ\tau)\circ \otimes :L\otimes L\to TL$; if the coequalizers exist in $C$ then we can form the quotient object $TL/I$ and there is an induced monoid structure in it. Under mild conditions on $C$, the natural morphism $i_L:L\to TL/I$ 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 $i_L:L\to TL/I$ is a monic morphism in $C$ and $U(L\coprod L)\cong U(L)\otimes U(L)$.

Properties

Isomorphism problem

Notice that the enveloping algebras of two Lie algebras may be isomorphic as associative algebras even if the two Lie algebras they arise from are not isomorphic to each other.

This happens even in characteristic zero, in which case, however, at least the dimension of the Lie algebra is encoded in its universal enveloping algebra, in the guise of the Gelfand-Kirillov dimension $GK\big(U(L)\big)$.

Poisson algebra structure on $U(\mathfrak{g})$

The universal enveloping algebra $U(\mathfrak{g})$ 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 $U(\mathfrak{g})$

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 $\epsilon:U(L)\to U(0)=\mathbf{1}$. The coproduct $\Delta:U(L)\to U(L\times L)\cong U(L)\otimes U(L)$ is induced by the diagonal map $L\to L\times L$ whereas the antipode $S=U(-id):U(L)\to U(L)$. One checks that these morphisms make $U(L)$ into a Hopf algebra in $C$. (e.g Milnor-Moore 65, section 5) The Milnor-Moore theorem states conditions under which the converse holds (hence under which a primitively generated Hopf algebra is a universal enveloping algebra of a Lie algebra).

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 characteristic zero is canonically isomorphic to a symmetric algebra $Sym(g)$, and $U(g)$ is isomorphic to $S(g)$ as a coalgebra, via the projection map $U(g)\to Gr U(g)$.

Relation to formal deformation quantization

See at formal deformation quantization the section Relation to universal enveloping algebras.

Relation to the group algebra

The universal enveloping algebra of a Lie algebra is the analogue of the usual group algebra of a group. It has the analogous function of exhibiting the category of Lie algebra modules as a category of modules for an associative algebra. This becomes more than an analogy when the universal enveloping algebra is viewed with its full Hopf algebra structure. By dualization, one obtains a commutative Hopf algebra which, in the case where the Lie algebra is that of an irreducible algebraic group over a field of characteristic 0, contains the algebra of polynomial functions of that group as a sub Hopf algebra in a natural fashion.

(quoted from Hochschild 1981, p. 221, see ibid. Thm. 3.1 on p. 230; Tjin 1992, Thm. 1)

Examples

Example

Consider the standard symplectic form on the Cartesian space $\mathbb{R}^{2n}$, making a symplectic vector space. This gives rise to the corresponding Heisenberg Lie algebra.

Depending on conventions, the universal enveloping algebra of the Heisenberg Lie algebra either already is the Weyl algebra on $2n$ generators or else it becomes so after after forming the quotient algebra in which the central generator is identified with the unit element of the ground field – whereas in the former case (considered eg. in Kravchenko 2000, Def. 2.1; Bekaert 2005, p. 9) the central generator plays the role of the formal Planck constant $\hbar$ with the Weyl algebra regarded as a formal deformation quantization of the symplectic manifold $\mathbb{R}^{2m}$.

Accordingly, given a Heisenberg Lie $n$-algebra it makes sense to call its universal enveloping $E_n$-algebra a Weyl $n$-algebra.

Example

(Universal enveloping of a tangent Lie algebra)
The universal enveloping algebra of the tangent Lie algebra of a finite-dimensional Lie group $G$ over the 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

• oidification to universal enveloping algebroid

• universal enveloping E-n algebra

References

• Jean-Pierre Serre: Universal Algebra of a Lie Algebra, Chapter III in: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [doi:10.1007/978-3-540-70634-2]

• Jacques Dixmier; Algèbres Enveloppantes, Cahiers Scientifique (1974), Engl transl: Enveloping Algebras, Graduate Studies in Mathematics 11 American Mathematical Society (1996) [ams:gsm-11]

• Nicolas Bourbaki, §I.2 in: Lie groups and Lie algebras – Chapters 1-3, Springer (1975, 1989) [ISBN:9783540642428]

• Gerhard P. Hochschild, The Universal Enveloping Algebra, Chapter XVI in: Basic Theory of Algebraic Groups and Lie Algebras, Graduate Texts in Mathematics 75, Springer (1981) [doi:10.1007/978-1-4613-8114-3_16]

Further discussion in the context of quantum groups:

• Tjark Tjin, An introduction to quantized Lie groups and algebras, Int. J. Mod. Phys. A 7 (1992) 6175-6213 [arXiv:hep-th/9111043, doi:10.1142/S0217751X92002805]

Discussion in the generality of super Lie algebras such as notably of the Whitehead Lie algebras arising in algebraic topology:

• John Milnor, John Moore, §5 in On the structure of Hopf algebras, Annals of Math. 81 (1965) 211-264 [doi:10.2307/1970615, pdf]

• Peter May, Kate Ponto, Def. 22.1.3 in: More concise algebraic topology – Localization, Completion, and Model Categories, University of Chicago Press (2012) [ISBN:9780226511795, pdf]

See also:

• 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.