nLab
universal enveloping algebra

Contents

Definition

For Lie algebras

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

For L 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 nn \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 CC admits countable coproducts, form the tensor algebra TL= n=0 L nTL=\coprod_{n=0}^\infty L^{\otimes n} on the object LL; this is a monoid in CC. In most standard cases, one can also form the smallest 2-sided ideal (i.e. AA-subbimodule) II in monoid AA among those ideals whose inclusion into AA is factorizing the map ([,]m TL+m TLτ):LLTL([,]-m_{TL}+m_{TL}\circ\tau)\circ \otimes :L\otimes L\to TL; if the coequalizers exist in CC then we can form the quotient object TL/ITL/I and there is an induced monoid structure in it. Under mild conditions on CC, the natural morphism i L:LTL/Ii_L:L\to TL/I is an universal enveloping monoid of LL in CC. If CC is an abelian tensor category and some flatness conditions on the tensor product are satisfied, then the enveloping monoid i L:LTL/Ii_L:L\to TL/I is a monic morphism in CC and U(LL)U(L)U(L)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 CC are isomorphic as associative monoids in CC, but this does not imply that the Lie algebras are isomorphic. This is even not true in general for the Lie kk-algebras (in classical sense), even if kk is a field of characteristics zero. It is known however in that case that the dimension of the finite-dimensional Lie kk-algebra LL can be read off from its universal enveloping kk-algebra as its Gel’fand-Kirillov dimension GK(U(L))GK(U(L)).

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

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

Suppose the universal enveloping algebras of Lie algebras exist in a kk-linear symmetric monoidal category CC and the functorial choice LU(L)L\mapsto U(L) realizing the above construction with tensor products is fixed. For example, this is true in the category of kk-modules where kk is a commutative ring. Then the projection L0L\to 0 where 00 is the trivial Lie algebra induces the counit ϵ:U(L)U(0)=1\epsilon:U(L)\to U(0)=\mathbf{1}. The coproduct Δ:U(L)U(LL)U(L)U(L)\Delta:U(L)\to U(L\coprod L)\cong U(L)\otimes U(L) is induced by the diagonal map LLLL\to L\coprod L whereas the antipode S=U(id):U(L)U(L)S=U(-id):U(L)\to U(L). One checks that these morphisms make U(L)U(L) into a Hopf algebra in CC.

If the category is simply the vector spaces over a field kk, then for lLl\in L, after we identify LL with its image in U(L)U(L), Δ(l)=l1+1l\Delta(l) = l\otimes 1 + 1\otimes l, i.e. the elements in LL are the primitive elements in U(L)U(L).

PBW theorem

The Poincaré–Birkhoff–Witt theorem states that the associated graded algebra of an enveloping algebra U(g)U(g) in characteristics zero is canonically isomorphic to a symmetric algebra Sym(g)Sym(g), and U(g)U(g) is isomorphic to S(g)S(g) as a coalgebra, via the projection map U(g)GrU(g)U(g)\to 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 GG over real or complex numbers is canonically isomorphic to the algebra of the left invariant differential operators on GG.

Literature

Revised on May 20, 2013 16:25:19 by Zoran Škoda (161.53.130.104)