On Weakly Complete Universal Enveloping Algebras:
A Poincaré–Birkhoff–Witt Theorem

In [7] we have initiated the theory of weakly complete universal enveloping algebras over $\mathbb{K}$ hoping that in some fashion this concept would resemble the classical universal enveloping algebra of a Lie algebra such as it is presented in the famous Poincaré-Birkhoff-Witt-Theorem (see e.g. [1], Chap. 1, Paragraph 2, n^o 7, Théorème 1., p.30). While this was not exactly the case, we shall discuss now how close we come to that theorem.

So we let $\mathbb{K}$ denote one of the topological fields $\mathbb{R}$ or $\mathbb{C}$. For a topological Lie algebra $\mathfrak{g}$ over $\mathbb{K}$ we let ${\cal I}(\mathfrak{g})$ denote the filter basis of all closed ideals $\mathfrak{i}\subseteq\mathfrak{g}$ such that $\dim\mathfrak{g}/\mathfrak{i}<\infty$.

A topological Lie algebra $\mathfrak{g}$ over $\mathbb{K}$ is called profinite-dimensional if $\mathfrak{g}=\lim_{\mathfrak{i}\in{\cal I}(\mathfrak{g})}\mathfrak{g}/\mathfrak{i}$. Let ${\cal W}\hskip-1.0pt{\cal L}$ denote the category of profinite-dimensional Lie algebras (over $\mathbb{K}$) and continuous Lie algebra morphisms between them.

Notice that by its definition every profinite-dimensional Lie algebra is weakly complete. A comment following Theorem 3.12 of [2] exhibits an example of a weakly complete $\mathbb{K}$-Lie algebra which is not a profinite-dimensional Lie algebra.

Let ${\cal W}\hskip-2.0pt{\cal A}$ denote the category of weakly complete associative unital algebras over $\mathbb{K}$. However, instead of considering the full category of weakly complete Lie algebras over $\mathbb{K}$, in the following we consider ${\cal W}\hskip-1.0pt{\cal L}$, the category of profinite-dimensional Lie algebras over $\mathbb{K}$ and continuous $\mathbb{K}$-Lie algebra morphisms. The reason for this restriction is Theorem 7.1 stating that every weakly complete unital $\mathbb{K}$-algebra is the projective limit of its finite-dimensional quotient algebras. This implies at once the following {Proposition} Let $A$ be any weakly complete unital $\mathbb{K}$-algebra and $A_{\rm Lie}$ the weakly complete Lie algebra obtained by considering on the weakly complete vector space $A$ the Lie algebra obtained with the Lie bracket $[x,y]=xy-yx$. Then $A_{\rm Lie}$ is profinite-dimensional.

The functor which associates with a weakly complete associative algebra $A$ the profinite-dimensional Lie algebra $A_{\rm Lie}$ is called the underlying Lie algebra functor.

For the complete proof of the following existence theorem, we shall invoke a considerable portion of a bulk category theoretical arguments. We shall collect these in an appendix, since, firstly, they reach far beyond the current application and, secondly, their full presentation might have led the reader astray from the present line of thought had we presented them at this point.

(The Existence Theorem of $\mathop{\bf U\hphantom{}}\nolimits$) The underlying Lie algebra functor $A\mapsto A_{\rm Lie}$ from ${\cal W}\hskip-2.0pt{\cal A}$ to ${\cal W}\hskip-1.0pt{\cal L}$ has a left adjoint $\mathop{\bf U\hphantom{}}\nolimits\colon{\cal W}\hskip-1.0pt{\cal L}\to{\cal W}\hskip-2.0pt{\cal A}$.

The front adjunction $\lambda_{\mathfrak{g}}\colon\mathfrak{g}\to\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})_{\rm Lie}$ is an embedding of profinite-dimensional Lie algebras.

The category ${\cal W}\hskip-1.0pt{\cal L}$ is complete. (Exercise. Cf. Theorem A3.48 of [11], p. 819.) The “Solution Set Condition” (of Definition A3.58 in [11], p. 824) holds. (Exercise: Cf. the proof Lemma 3.58 of [11], p. 91.) Hence $\mathop{\bf U\hphantom{}}\nolimits$ exists by the Adjoint Functor Existence Theorem (i.e., Theorem A3.60 of [11], p. 825).

The assertion about $\lambda_{\mathfrak{g}}$ being an embedding follows from Proposition 7.5 (ii) in the Appendix.

In other words, each profinite-dimensional Lie algebra $\mathfrak{g}$ may be considered as a closed Lie subalgebra of $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})_{\rm Lie}$ with the property that each continuous Lie algebra morphism $f\colon\mathfrak{g}\to A_{\rm Lie}$ for some weakly complete associative unital algebra $A$ extends uniquely to a ${\cal W}\hskip-2.0pt{\cal A}$-morphism $f^{\prime}\colon\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})\to A$.

For each profinite-dimensional $\mathbb{K}$-Lie algebra, we shall call $\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{K}}(\mathfrak{g})$ the weakly complete enveloping algebra of $\mathfrak{g}$ (over $\mathbb{K}$).

For every profinite-dimensional Lie algebra $\mathfrak{g}$, a morphism $f\colon\mathfrak{g}\to\mathbb{K}$, $f(x)=0$, according to the definition of $\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{K}}(\mathfrak{g})$ induces a natural ${\cal W}\hskip-2.0pt{\cal A}$-morphism $\alpha_{\mathfrak{g}}\colon\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{K}}(\mathfrak{g})\to\mathbb{K}$ such that $\alpha_{\mathfrak{g}}(\mathfrak{g})=\{0\}$ and that $\alpha_{\mathfrak{g}}\circ\iota_{{\mathop{\bf U\hphantom{}}\nolimits}(\mathfrak{g})}=\mathop{\rm id}\nolimits_{\mathbb{K}}$.

The retraction $\alpha_{\mathfrak{g}}$ is also called the augmentation of $\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{K}}(\mathfrak{g})$.

In the Appendix we shall also introduce for any weakly complete vector space $W$ its weakly complete tensor algebra $\mathop{\bf T\hphantom{}}\nolimits(W)$ (cf. paragraph (C) preceding Proposition 7.5 and Theorem 8) and show that $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ is a quotient algebra of $\mathop{\bf T\hphantom{}}\nolimits(|\mathfrak{g}|)$ if $|\mathfrak{g}|$ is the underlying weakly complete vector space underlying $\mathfrak{g}$. There is a commutative diagram

Moreover, we have the following corollary to our existence theorem:

For any profinite-dimensional Lie algebra $\mathfrak{g}$, the unital associative subalgebra $\langle\mathfrak{g}\rangle$ generated algebraically in $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ by $\mathfrak{g}$ is dense in $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$.

The assertion follows from Proposition 7.5 in the Appendix.

Of course we would like to have a better insight into the structure of the algebra $\langle\mathfrak{g}\rangle$. This information we provide in the following section and thereby close the gap between the concepts of the weakly complete enveloping algebra and the classical universal enveloping abstract algebra dealt with in the Poincaré-Birkhoff-Witt Theorem.

We briefly recall that the functor which assigns to a unital $\mathbb{K}$-algebra $X$ the underlying Lie algebra $X_{\rm Lie}$ (with the underlying $\mathbb{K}$ vector space of $X$ as vector space structure endowed with the bracket operation $(x,y)\mapsto[x,y]:=xy-yx$ as Lie bracket) has a left adjoint functor $U$ which assigns to a Lie algebra $L$ a unital associative algebra $U(L)$ and a natural Lie algebra morphism $\rho_{L}\colon L\to U(L)_{\rm Lie}$ such that for each Lie algebra morphim $f\colon L\to X_{\rm Lie}$ for a unital algebra $X$ there is a unique morphism of unital algebras $f^{\prime}\colon U(L)\to X$ such that $f=f^{\prime}_{\rm Lie}\circ\rho_{L}$. The algebra $U(L)$ is called the universal enveloping algebra of $L$. A large body of text book literature is available on it. A prominent result is the Poincaré-Birkhoff-Witt Theorem on the structure of $U(L)$ which implies in particular that $\rho_{L}\colon L\to U(L)_{\rm Lie}$ is injective.

From the Theorem of Poincaré, Birkhoff and Witt it is known that $\rho_{L}$ is injective. One may therefore assume that $L\subseteq U(L)$ such that $\rho_{L}$ is the inclusion function. (See [1] or [3].) In this parlance the universal property reads as follows:

Also from the Theorem of Poincaré, Birkhoff and Witt we know that

In the present section we shall now denote by $|\mathfrak{g}|$ the abstract Lie algebra underlying the profinite-dimensional Lie algebra $\mathfrak{g}$. Then the main result of this section will be a complete clarification of the relation of the weakly complete enveloping algebra $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ of a profinite-dimensional Lie algebra $\mathfrak{g}$ and the universal enveloping algebra $U(|\mathfrak{g}|)$ of $|\mathfrak{g}|$.

For a profinite-dimensional Lie algebra $\mathfrak{g}$ there is a natural morphism $\varepsilon_{\mathfrak{g}}\colon U(|\mathfrak{g}|)\to|\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})|$ of unital algebras such that

1. (i)

   the following diagram is commutative:

   $$\begin{matrix}|\mathfrak{g}|&\smash{\mathop{\hbox to40.0pt{\rightarrowfill}}\limits^{\mathop{\rm incl}\nolimits}}&U(|\mathfrak{g}|)\\ \hbox to0.0pt{\hss$\vbox{\hbox{$\scriptstyle\mathop{\rm id}\nolimits_{|\mathfrak{g}|}$}}$}\Big{\downarrow}&&\Big{\downarrow}\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle\varepsilon_{\mathfrak{g}}$}}$\hss}\\ |\mathfrak{g}|&\smash{\mathop{\hbox to40.0pt{\rightarrowfill}}\limits_{|\mathop{\rm incl}\nolimits|}}&|\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})|.\end{matrix}$$

2. (ii)

   The image of $\varepsilon_{\mathfrak{g}}$ is dense in $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$.

3. (iii)

   The morphism $\varepsilon_{\mathfrak{g}}$ is injective if $\mathfrak{g}$ is finite-dimensional.

(i) The claim is a direct consequence of the universal property of the functor $U$.

(ii) We have $\mathop{\rm im}\nolimits(\varepsilon_{\mathfrak{g}})=\varepsilon_{\mathfrak{g}}(U(|\mathfrak{g}|))=\varepsilon_{\mathfrak{g}}(\langle|\mathfrak{g}|\rangle=\langle\varepsilon_{\mathfrak{g}}(\mathfrak{g})\rangle=\langle|\mathfrak{g}|\rangle$ in $|\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})|$. From Corollary 1 we know that $\langle|\mathfrak{g}|\rangle$ is dense in $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$.

(iii) If $\mathfrak{g}$ is finite-dimensional, then every finite-dimensional Lie algebra representation $\rho\colon\mathfrak{g}\to A_{\rm Lie}=\mathrm{End}(V)_{\rm Lie}$ for a finite-dimensional vector space $V$ extends to an associative representation $\rho^{\prime}\colon\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})\to\mathrm{End}(V)$ . Then $\rho\circ\varepsilon_{\mathfrak{g}}\colon U(\mathfrak{g})\to\mathrm{End}(V)$ is an extension to an associative representation of $U(\mathfrak{g})$ which is unique. By Harish-Chandra’s Lemma (see Dixmier [3], 2.5.7), the extensions of associative representations of $U(\mathfrak{g})$ of finite-dimensional Lie algebra representations of $\mathfrak{g}$ separate the points of $U(\mathfrak{g})$ and so the claim follows.

The remainder of this section now is devoted to removing the restriction to finite-dimensionality in Lemma 2(iii). That is, we want to show

For a profinite-dimensional Lie algebra $\mathfrak{g}$, the algebra morphism $\varepsilon_{\mathfrak{g}}\colon U(|\mathfrak{g}|)\to|\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})|$ is injective.

The proof will occupy the remainder of this section. We shall resort to the existing literature on $U(L)$ such as [1] or [3]. We are given the profinite-dimensional Lie algebra $\mathfrak{g}$ and we write $L:=|\mathfrak{g}|$ for the underlying Lie algebra. So $L\subseteq U(L)_{\rm Lie}$. Let $B$ be a totally ordered basis of $L$. We begin by recalling the following basic fact from the Poincaré-Birkhoff-Witt Theorem (see [1], Corollary 3, Section 7 of Paragraph 2):

Now assume that

Then there is a finite subset $F\subseteq B$ such that $u\in\mathop{\rm span}\nolimits(\widetilde{F})$ for

There is a closed ideal $J$ of $\mathfrak{g}$ so that $J\cap\mathop{\rm span}\nolimits F=\{0\}$. {Proof} The vector space $V=\mathop{\rm span}\nolimits F$ is finite-dimensional. Let $C$ be the boundary of a compact 0-neighborhood in $V$. Then $V=\mathbb{K}{\cdot}C$.

Returning at this point to the fact that $\mathfrak{g}$ is a profinite-dimensional Lie algebra, we conclude that there is a filterbasis ${\cal I}$ of closed ideals $I$ of $\mathfrak{g}$ such that $\dim\mathfrak{g}/I<\infty$ for $I\in{\cal I}$ such that $\mathfrak{g}\cong\lim_{I\in{\cal I}}\mathfrak{g}/I$. In particular, $\bigcap{\cal I}=\{0\}$. Therefore $\bigcap_{I\in{\cal I}}(C\cap I)=C\cap\{0\}=\emptyset$. Since $C$ is compact, the filter basis $\{C\cap I:I\in{\cal I}\}$ with empty intersection consists of compact sets and therefore must contain the empty set. Thus there is a $J\in{\cal I}$ such that $C\cap J=\emptyset$. In fact, we have $J\cap\mathop{\rm span}\nolimits F=\{0\}$. Indeed, suppose there were a nonzero $t\in\mathbb{K}$ and a $c\in C$ such that $t{\cdot}c\in J$, then $c=t^{-1}{\cdot}(t{\cdot}c)\in t^{-1}{\cdot}J=J$ which is impossible.

Assume that there is an ideal $J$ of $L$ such that $J\cap\mathop{\rm span}\nolimits F=\{0\}$. Then the image of $u$ under the morphism $U(L)\to U(L/J)$ is nonzero.

We choose a finite dimensional vector subspace $H$ of $L$ containing $F$ such that $L=H\oplus J$. Let $E$ be a totally ordered basis for $H$ such that the order of $E$ extends that of $F$, choose a totally ordered basis $D$ of $J$ and make sure that $E\cup D$ has a total order extending the orders of $E$ and $D$, thus yielding a totally ordered basis of $L$.

We consider the quotient morphism of Lie algebras $q_{J}\colon L\to L/J$. Then $q_{J}$ maps $E$ bijectively onto a basis $E^{\prime}=q_{J}(E)$ of $L/J$. Let

Then $U(q_{J})\colon U(L)\to U(L/J)$ maps $\widetilde{E}$ bijectively onto a basis $\widetilde{E^{\prime}}$ of $U(L/J)$ by (PBW). If we now write $u=\sum_{S\in\widetilde{E}}c_{S}{\cdot}S$ with $c_{S}\in\mathbb{K}$, then $U(q_{J})(u)=\sum_{q_{J}(S)\in\widetilde{E^{\prime}}}c_{S}{\cdot}q_{J}(S)\neq 0$, since $\widetilde{E^{\prime}}$ is a basis of $U(L/J)$.

As the kernel of the quotient map $U(L)\to U(L/J)$ is $U(L)J$, the claim of the preceding lemma may be expressed equivalently in the form $u\notin U(L)J$.

Now we recall that $q_{J}\colon L\to L/J\subseteq U(L/J)$ is in fact the underlying Lie algebra morphism $|p_{J}|$ of a quotient morphism $p_{J}\colon\mathfrak{g}\to\mathfrak{g}/J$ of profinite Lie algebras and that $q_{J}$ extends uniquely to an algebra morphism $U(|p_{J}|)\colon U(L)\to U(L/J)$ with kernel $U(L)J$. Then from Lemma 2 we know that $U(|p_{J}|)(u)$ is nonzero in $U(L/J)$ and from Lemma 2 we infer that $\varepsilon_{\mathfrak{g}/J}$ is injective. The commutative diagram

then shows that $\varepsilon_{\mathfrak{g}}(u)\neq 0$. Therefore, since $u\in U(L)\setminus\{0\}$ was arbitrary, $\varepsilon_{\mathfrak{g}}$ is injective, leaving the elements of $L=|\mathfrak{g}|$ fixed. This completes the proof of Lemma 2. Thus $U(|\mathfrak{g}|)$ may be considered as a subalgebra of $|\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})|$, containing $|\mathfrak{g}|\subseteq|\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})|$.

This may be rephrased in the following Theorem which summarizes our efforts to elucidate the close relation between $U(|\mathfrak{g}|)$ and $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$:

(The Relation of $U(-)$ and $\mathop{\bf U\hphantom{}}\nolimits(-)$) For any profinite-dimensional real or complex Lie algebra $\mathfrak{g}$ considered as a closed Lie subalgebra of $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})_{\rm Lie}$, the associative unital subalgebra $\langle\mathfrak{g}\rangle$ generated algebraically by $\mathfrak{g}$ in $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ is naturally isomorphic to $U(|\mathfrak{g}|)$ (under an isomorphism fixing the elements of $\mathfrak{g}$) and is dense in $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$.

In a slightly careless sense we may memorize this as saying:

Since every weakly complete unital algebra is a strict projective limit of all finite-dimensional quotient algebras, it will now turn out to be sufficient to test the universal property of the functor $\mathop{\bf U\hphantom{}}\nolimits$ only for finite-dimensional unital associative algebras:

Assume that the profinite-dimensional Lie algebra $\mathfrak{g}$ is contained functorially in a weakly complete unital algebra $\mathop{\bf V\hphantom{}}\nolimits(\mathfrak{g})$ such that for each finite-dimensional unital algebra $A$ and each morphism of profinite-dimensional Lie algebras $f\colon\mathfrak{g}\to A_{\rm Lie}$ there is a unique morphism of weakly complete unital algebras $f^{\prime}\colon\mathop{\bf V\hphantom{}}\nolimits(\mathfrak{g})\to A$ extending $f$. Then $\mathop{\bf V\hphantom{}}\nolimits(\mathfrak{g})\cong\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ naturally.

We apply the Density and Adjunction Theorem 7.2 in the Appendix with ${\cal A}$ as the category of weakly complete associative unital algebras, and ${\cal B}$ as the category of profinite-dimensional Lie algebras with the full subcategory ${\cal B}_{d}$ of finite-dimensional Lie algebras which is topologically dense in ${\cal B}$. Then, by hypothesis, the function $\mathop{\bf V\hphantom{}}\nolimits\colon\mathop{\rm ob}\nolimits{\cal B}\to\mathop{\rm ob}\nolimits{\cal A}$ is conditionally left adjoint to the functor $(\cdot)_{\rm Lie}\colon{\cal A}\to{\cal B}$ which maps an associative algebra to the Lie algebra with the Lie bracket $[a,b]=ab-ba$ (see Definition 7.2). Then by Theorem 7.2, $\mathop{\bf V\hphantom{}}\nolimits$ is naturally isomorphic to the left adjoint $\mathop{\bf U\hphantom{}}\nolimits$ of $(\cdot)_{\rm Lie}$.

Perhaps more deeply we shall see now that, while as a left-adjoint functor, $\mathop{\bf U\hphantom{}}\nolimits$ preserves colimits, is also preserve certain limits, namely, the projective limits $\mathfrak{g}=\lim_{\mathfrak{i}\in{\cal I}(\mathfrak{g})}\mathfrak{g}/\mathfrak{i}$ of Definition 1. Indeed in Theorem 7.5 in the Appendix we show:

($\mathop{\bf U\hphantom{}}\nolimits$ preserves some projective limits) For a profinite-dimensional Lie algebra $\mathfrak{g}$ with its filter basis ${\cal I}(\mathfrak{g})$ of cofinite-dimensional ideals $\mathfrak{i}$ we have

The argument in the Appendix shows, that while the assertion of the theorem is natural and easy to absorb, its proof is deeper than one would expect initially.

We now address the important aspect of enveloping algebras from their beginning, namely, the fact that they are symmetric Hopf algebras. For some of the proofs in this section we refer to our predecessor paper [7].

The universal enveloping functor $\mathop{\bf U\hphantom{}}\nolimits$ is multiplicative, that is, there is a natural isomorphism $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g}_{1}\times\mathfrak{g}_{2})\to\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g}_{1})\otimes_{\cal W}\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g}_{2})$.

For any weakly complete unital algebra $A$, the vector space morphism $\Delta_{A}\colon A\to A\otimes_{\cal W}A$, $\Delta_{A}(a)=a\otimes 1+1\otimes a$ is a morphism of weakly complete Lie algebras $A_{\rm Lie}\to(A\otimes_{W}A)_{\rm Lie}$.

$p_{\mathfrak{g}}=\delta_{\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})}\circ\lambda_{\mathfrak{g}}\colon\mathfrak{g}\to(\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})\otimes_{\cal W}\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g}))_{\rm Lie}$

Now we have

($\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ as a Hopf algebra)

For (a), see [7], Corollary 6.5.

For (b) we consider a morphism $f\colon\mathfrak{g}\to\mathfrak{h}$ a morphism of profinite-dimensional Lie algebras and for the functoriality of $\mathop{\bf U\hphantom{}}\nolimits$ (short for $\mathop{\bf U\hphantom{}}\nolimits_{K}(-)$) as regards to comultiplication $\gamma_{\mathfrak{g}}\colon\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})\to\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})\otimes\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ we verify the commutativity of the following diagram (with $\otimes=\otimes_{\cal W}$)

(See also Proposition 4.) Coidentity and symmetry are treated similarly.

This proposition expresses the fact that $\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{K}}$ is a functor from the category of profinite-dimensional Lie algebras to the category of weakly complete symmetric Hopf algebras. Its significance is emphasised by the fact that essential portions of the noteworthy theory of weakly complete symmetric Hopf algebras have meanwhile entered the textbook literature. (See [11], Appendix A3, Appendix A7, Chapter 3–Part 3.) We have collected some essential features in our Appendix such as Theorems 9 and 9.

Now we specialize these to the case of $A=\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{K}}(\mathfrak{g})$. We use the notation $\mathbb{R}_{<}=\{r\in\mathbb{R}:0<r\}$ and recall that $\mathfrak{g}\subseteq A$ and that $A^{\times}$ denotes the group of units of $A$. For the exponential function $\exp\colon A_{\rm Lie}\to A^{\times}$ as in Theorem 9 we define the closed subgroup

The following theorem now is a principal result in the theory of weakly complete enveloping algebras of profinite-dimensional real or complex Lie algebras.

(The Weakly Complete Enveloping Hopf Algebra) Let $\mathfrak{g}$ be a profinite-dimensional Lie algebra and $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ its weakly complete enveloping algebra containing $\mathfrak{g}$ according to Theorem 2. Then the following statements hold:

1. (a)

   The group of units $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})^{\times}$ is dense in $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$. It is an almost connected pro-Lie group, connected in the case of $\mathbb{K}=\mathbb{C}$. The algebra $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ has an exponential function $\exp\colon{\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})_{\rm Lie}}\to\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})^{\times}$. The Lie algebra $\mathfrak{L}(\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})^{\times})$ of $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})^{\times}$ is (naturally isomorphic to) $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})_{\rm Lie}$.

2. (b)

   The pro-Lie algebra $\mathbb{P}(\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g}))$ is the Lie algebra of the pro-Lie group $\mathbb{G}(\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g}))$ of grouplike elements and the restriction and corestriction of $\exp$ is the exponential function for this group.

3. (c)

   The profinite-dimensional Lie algebra $\mathbb{P}(\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g}))$ contains

   $$\mathfrak{g}=\mathbb{P}(U(|\mathfrak{g}|))=\mathbb{P}(\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g}))\cap U(|\mathfrak{g}|).$$

4. (d)

   For $\mathbb{K}=\mathbb{R}$, the restriction and corestriction of $\exp$ yields the exponential function

   $$\exp_{\Gamma^{*}(\mathfrak{g})}\colon\mathfrak{g}=\mathfrak{L}(\Gamma^{*}(\mathfrak{g}))\to\Gamma^{*}(\mathfrak{g})$$

   of that pro-Lie subgroup $\Gamma^{*}(\mathfrak{g})$ of $\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{R}}(\mathfrak{g})^{\times}$ whose Lie algebra is precisely $\mathfrak{g}$.

5. (e)

   Define the hyperplane ideal ${\mathbb{I}}$ as the kernel of the augmentation $\alpha_{\mathfrak{g}}$. Then we have

   1. (i)

      for $\mathbb{K}=\mathbb{R}$: $\exp(\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{R}}(\mathfrak{g}))=(\mathbb{R}_{<}{\cdot}1)\oplus{\mathbb{I}}$, an open half space,

   2. (ii)

      for $\mathbb{K}=\mathbb{C}$: $\exp(\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{C}}(\mathfrak{g}))=\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{C}}(\mathfrak{g})\setminus{\mathbb{I}}$.

For the proofs of (a) and (b) see Theorem 9 in the Appendix.

The proof of (c) follows from [7], Theorem 3.4 and Theorem 2. Cf. also [11], Theorem A3.102 and its proof for $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$.

The proof of (d) must verify that $\mathfrak{L}(\overline{\langle\mathfrak{g}\rangle})\subseteq\mathfrak{g}$. This conclusion we derive from [10], Corollary 4.22 and its proof.

The proof of (e) follows from Theorem 9 in the Appendix.

Here are some immediate consequences:

The weakly complete enveloping algebra $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ of a nonzero profinite-dimensional weakly complete Lie algebra $\mathfrak{g}$ has nontrivial grouplike elements contained in $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})^{\times}$. Specifically, there is a pro-Lie subgroup $\Gamma^{*}(\mathfrak{g})$ of grouplike elements whose Lie algebra is isomorphic to $\mathfrak{g}$ and whose exponential function is induced by that of $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$.

By contrast, on the purely algebraic side, the universal enveloping Hopf algebra $U(L)$ of a Lie algebra $L$ shows no visible nontrivial grouplike elements while a nontrivial weakly complete enveloping algebra always does.

For any profinite-dimensional Lie algebra $\mathfrak{g}$ there is a pro-Lie group $G$ whose Lie algebra $\mathfrak{L}(G)$ may be identified with $\mathfrak{g}$.

Indeed the theorem provides a weakly complete unital algebra $A$ with an exponential function $\exp_{A}$ such that $\exp_{G}\colon\mathfrak{L}(G)\to G$ may be identified with a restriction and corestriction of $\exp_{A}$.

This is indeed much more than what is historically known as Sophus Lie’s Third Fundamental Theorem.

For an abelian Lie algebra $\mathfrak{g}$, the weakly complete unital algebra $\mathop{\bf U\hphantom{}}\nolimits(\mathfrak{g})$ is commutative. In various special aspects we considered this situation in [2], Lemmas 3.4, 3.5, and 3.10ff., and in [7], Section 5 and Example 6.2. We now return to the commutative situation more systematically now and discuss the structure of $\mathop{\bf U\hphantom{}}\nolimits_{\mathbb{K}}(\mathfrak{g})$ completely for $\dim\mathfrak{g}=1$, and derive consequences for the abelian case in general.