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 , 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 , its primitive elements are those elements satisfying

where 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 is fully faithful; that is, there is a natural bijection between Lie algebra homomorphisms and Hopf algebra homomorphisms . This is more or less equivalent to the claim that the natural inclusion induces an isomorphism from to the Lie algebra of primitive elements of , 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).

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Drawing subgroups of the modular group

Previously we learned how to count the finite index subgroups of the modular group . 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.

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Connected components in a distributive category

Previously we claimed that if you want to check whether a category “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 be a distributive category with terminal object , and let be the coproduct of two copies of . For an object , what does look like? If and is a sufficiently well-behaved topological space, morphisms correspond to subsets of the connected components of , and naturally has have the structure of a Boolean algebra or Boolean ring whose elements can be interpreted as subsets of the connected components of .

It turns out that 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 the object this functor outputs is known as the Pierce spectrum.

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

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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 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 with finite products and finite coproducts such that finite products distribute over finite coproducts; more explicitly, the natural maps

should be isomorphisms, and also the natural maps should be isomorphisms, where 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:

• 
• More generally, any bicartesian closed category, and in particular any topos
• 
• 

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

• 
• 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 is distributive!

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The Lawvere theory of Boolean functions

Let be a set with two elements. The category of Boolean functions is the category whose objects are the finite powers of 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 , it is a Lawvere theory.

Question: What are models of this Lawvere theory?

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Conjugacy classes of finite index subgroups

Previously we learned how to count the number of finite index subgroups of a finitely generated group . 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 .

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 is a groupoid, then , the free loop space or inertia groupoid of , is the groupoid of maps , where  is the groupoid  with one object and automorphism group . Explicitly, this groupoid has

• objects given by automorphisms  of the objects , and
• morphisms  given by morphisms in  such that

.

It’s not hard to see that , so to understand this construction for arbitrary groupoids it’s enough to understand it for connected groupoids, or (up to equivalence) for groupoids with a single object and automorphism group . In this case, is the groupoid with objects the elements of and morphisms given by conjugation by elements of ; equivalently, it is the homotopy quotient or action groupoid of the action of on itself by conjugation.

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

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

I promise this is relevant to counting subgroups!

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We apologize for the interruption in service

My MaBloWriMo 2015 run met an untimely end on the 18th, when LaTeX stopped working on WordPress for me; I could no longer see any of the LaTex I was writing. It’s still not working for me in Chrome, but it’s now working for me in another browser, so hopefully I’ll get some posts up soon.

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Forms and Galois cohomology

Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence

where  is a Galois extension, is the Galois group of (thinking of as an object of the category of field extensions of at all times), is a category of “objects over ,” and is a category of “objects over .”

In fact this description is probably only correct if  is a finite Galois extension; if is infinite it should probably be modified by requiring that every function of that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on . To avoid this difficulty we’ll stick to the case that is a finite extension.

Today we’ll recover from this abstract description the somewhat more concrete punchline that -forms of an object can be classified by Galois cohomology , and we’ll give some examples.

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Stating Galois descent

After a relaxing and enjoyable break, we’re finally in a position to state what it means for structures to satisfy Galois descent.

Fix a field . The gadgets we want to study assign to each separable extension a category of “objects over ,” to each morphism of extensions an “extension of scalars” functor , and to each composable pair of morphisms of extensions a natural isomorphism

of functors (where again we’re taking compositions in diagrammatic order) satisfying the usual cocycle condition that the two natural isomorphisms we can write down from this data agree. We’ll also want unit isomorphisms satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions of to the 2-category , and in particular each naturally acquires an action of (where we mean automorphisms of extensions of , hence if is Galois this is the Galois group) in precisely the sense we described earlier.

We’ll call such an object a Galois prestack (of categories, over ) for short. The basic example is the Galois prestack of vector spaces , which sends an extension to the category of -vector spaces and sends a morphism to the extension of scalars functor

.

Every example we consider will in some sense be an elaboration on this example in that it will ultimately be built out of vector spaces with extra structure, e.g. the Galois prestacks of commutative algebras, associative algebras, Lie algebras, and even schemes. In these examples, fields are not really the natural level of generality, and to make contact with algebraic geometry we should replace them with commutative rings, but for now we’ll ignore this.

In order to state the definition, we need to know that if is an extension, then the functor naturally factors through the category of homotopy fixed points for the action of on . We’ll elaborate on why this is in a moment.

Definition: A Galois prestack satisfies Galois descent, or is a Galois stack, if for every Galois extension the natural functor (where ) is an equivalence of categories.

In words, this condition says that the category of objects over is equivalent to the category of objects over equipped with homotopy fixed point structure for the action of the Galois group (or Galois descent data).

(Edit, 11/18/15:) This definition is slightly incorrect in the case of infinite Galois extensions; see the next post and its comments for some discussion.

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Finite index subgroups of the modular group

Two weeks ago we proved the following formula. Let be a finitely generated group and let be the number of subgroups of of index . Then

.

This identity reflects, in a way we made precise in the previous post, the decomposition of a finite -set (the terms on the LHS) into a disjoint union of transitive -sets (the terms on the RHS).

Noam Zeilberger commented on the previous post that he had seen results like this for more specific groups in the literature; in particular, Samuel Vidal describes a version of this analysis for , the modular group. In this post we’ll use the above formula to compute the number of subgroups of index in using a computer algebra system that can manipulate power series. We’ll also say something about how to visualize these subgroups.

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