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## Lie algebras are groups

Once upon a time I imagine people were very happy to think of Lie algebras as “infinitesimal groups,” but presumably when infinitesimals fell out of favor this interpretation did too. In this post I’ll record an observation that can justify thinking of Lie algebras as groups in a strong sense: they are group objects in a certain category which can be interpreted as a category of “infinitesimal spaces.”

Below we work throughout over a field of characteristic zero.

For starters, the universal enveloping algebra functor $\mathfrak{g} \mapsto U(\mathfrak{g})$, which a priori takes values in algebras (it’s left adjoint to the forgetful functor from algebras to Lie algebras), in fact takes values in Hopf algebras. This upgraded functor continues to be a left adjoint, although the forgetful functor is less obvious. Given a Hopf algebra $H$, its primitive elements are those elements $x \in H$ satisfying

$\Delta x = x \otimes 1 + 1 \otimes x$

where $\Delta$ is the comultiplication. The primitive elements of a Hopf algebra form a Lie algebra, and this gives a forgetful functor from Hopf algebras to Lie algebras whose left adjoint is the upgraded universal enveloping algebra functor.

The key observation is that this upgraded functor $\mathfrak{g} \to U(\mathfrak{g})$ is fully faithful; that is, there is a natural bijection between Lie algebra homomorphisms $\mathfrak{g} \to \mathfrak{h}$ and Hopf algebra homomorphisms $U(\mathfrak{g}) \to U(\mathfrak{h})$. This is more or less equivalent to the claim that the natural inclusion $\mathfrak{g} \to U(\mathfrak{g})$ induces an isomorphism from $\mathfrak{g}$ to the Lie algebra of primitive elements of $U(\mathfrak{g})$, which can be proven using the PBW theorem.

Hence Lie algebras embed as a full subcategory of Hopf algebras; that is, they can be thought of as Hopf algebras satisfying certain properties, rather than having extra structure (in the nLab sense). What are these properties? For starters, they are all cocommutative. This is important because cocommutative Hopf algebras are group objects in the category of cocommutative coalgebras (this is not true with “cocommutative” dropped!), which in turn can be interpreted as a category of infinitesimal spaces. (For example, this category is cartesian closed, and in particular distributive.)

Hence Lie algebras are group objects in cocommutative coalgebras satisfying some property (for example, “conilpotence”; see Theorem 3.8.1 here).

## Drawing subgroups of the modular group

Previously we learned how to count the finite index subgroups of the modular group $\Gamma = PSL_2(\mathbb{Z})$. The worst thing about that post was that it didn’t include any pictures of these subgroups. Today we’ll fix that.

The pictures in this post can be interpreted in at least two ways. On the one hand, they are graphs of groups in the sense of Bass-Serre theory, and on the other hand, they are also dessin d’enfants (for the rest of this post abbreviated to “dessins”) in the sense of Grothendieck. But you don’t need to know that to draw and appreciate them.

## Connected components in a distributive category

Previously we claimed that if you want to check whether a category $C$ “behaves like a category of spaces,” you can try checking whether it’s distributive. The goal of today’s post is to justify the assertion that objects in distributive categories behave like spaces by showing that they have a notion of “connected components.”

For starters, let $C$ be a distributive category with terminal object $1$, and let $2 = 1 + 1$ be the coproduct of two copies of $1$. For an object $X \in C$, what does $\text{Hom}(X, 2)$ look like? If $C = \text{Top}$ and $X$ is a sufficiently well-behaved topological space, morphisms $X \to 2$ correspond to subsets of the connected components of $X$, and $\text{Hom}(X, 2)$ naturally has have the structure of a Boolean algebra or Boolean ring whose elements can be interpreted as subsets of the connected components of $X$.

It turns out that $\text{Hom}(X, 2)$ naturally has the structure of a Boolean algebra or Boolean ring (more invariantly, the structure of a model of the Lawvere theory of Boolean functions) in any distributive category. Hence any distributive category naturally admits a contravariant functor into Boolean rings, or, via Stone duality, a covariant functor into profinite sets / Stone spaces. This is our “connected components” functor. When $C = \text{Aff}$ the object this functor outputs is known as the Pierce spectrum.

This construction can be thought of as trying to do for $\pi_0$ what the étale fundamental group does for $\pi_1$.

## Distributive categories

Among all of the standard algebraic structures that a student typically encounters in an introduction to abstract algebra (groups, rings, fields, modules), commutative rings are somehow special: the opposite category $\text{CRing}^{op}$ behaves like a category of spaces, so much so that an entire field of mathematics is dedicated to doing geometry based on it.

In general, suppose you find yourself in some category. What sort of behavior could you look for that might qualify as “behaving like a category of spaces”?

One thing to look for is distributivity. Recall that a distributive category is a category $C$ with finite products $\times$ and finite coproducts $+$ such that finite products distribute over finite coproducts; more explicitly, the natural maps

$X \times Y+ X \times Z \to X \times (Y + Z)$

should be isomorphisms, and also the natural maps $0 \to X \times 0$ should be isomorphisms, where $0$ denotes the initial object. (Curiously, distributive categories are themselves like categorified versions of commutative rings.)

This is a pretty good test. The following familiar categories are distributive:

• $\text{Set}$
• More generally, any bicartesian closed category, and in particular any topos
• $\text{Top}$
• $\text{Aff} = \text{CRing}^{op}$

These are all reasonable candidates for categories of “spaces.” On the other hand, the following familiar categories are not distributive:

• $\text{Grp}$
• More generally, any nontrivial category with a zero object, and in particular any abelian category

You might object that there is also an entire field of mathematics dedicated to treating groups as geometric objects. I contend that the geometric object a group describes is actually a groupoid, and $\text{Gpd}$ is distributive!

## The Lawvere theory of Boolean functions

Let $2$ be a set with two elements. The category of Boolean functions is the category whose objects are the finite powers $2^k, k \in \mathbb{Z}_{\ge 0}$ of $2$ and whose morphisms are all functions between these sets. For a computer scientist, the morphisms of this category have the interpretation of functions which input and output finite sequences of bits.

Since this category has finite products and is freely generated under finite products by a single object, namely $2$, it is a Lawvere theory.

Question: What are models of this Lawvere theory?

## Conjugacy classes of finite index subgroups

Previously we learned how to count the number of finite index subgroups of a finitely generated group $G$. But for various purposes we might instead want to count conjugacy classes of finite index subgroups, e.g. if we wanted to count isomorphism classes of connected covers of a connected space with fundamental group $Gi$.

There is also a generating function we can write down that addresses this question, although it gives the answer less directly. It can be derived starting from the following construction. If $X$ is a groupoid, then $LX = [S^1, LX]$, the free loop space or inertia groupoid of $X$, is the groupoid of maps $S^1 \to X$, where $S^1$ is the groupoid $B\mathbb{Z}$ with one object and automorphism group $\mathbb{Z}$. Explicitly, this groupoid has

• objects given by automorphisms $f : x \to x$ of the objects $x \in X$, and
• morphisms $(f_1 : x_1 \to x_1) \to (f_2 : x_2 \to x_2)$ given by morphisms $g : x_1 \to x_2$ in $X$ such that

$x_1 \xrightarrow{f_1} x_1 \xrightarrow{g} x_2 = x_1 \xrightarrow{g} x_2 \xrightarrow{f_2} x_2$.

It’s not hard to see that $L(X \coprod Y) \cong LX \coprod LY$, so to understand this construction for arbitrary groupoids it’s enough to understand it for connected groupoids, or (up to equivalence) for groupoids $X = BG$ with a single object and automorphism group $G$. In this case, $LBG$ is the groupoid with objects the elements of $G$ and morphisms given by conjugation by elements of $G$; equivalently, it is the homotopy quotient or action groupoid of the action of $G$ on itself by conjugation.

In particular, when $G$ is finite, this quotient always has groupoid cardinality $1$. Hence:

Observation: If $X$ is an essentially finite groupoid (equivalent to a groupoid with finitely many objects and morphisms), then the groupoid cardinality of $LX$ is the number of isomorphism classes of objects in $X$.

I promise this is relevant to counting subgroups!