Posted by: yanzhang | October 2, 2009

A Free Association On Basic Adjoints

I’ve been betraying the title of this blog and doing some abstract nonsense lately, mostly to relearn algebraic geometry for the n-th time for some embarrassingly large n. With the memory capabilities of a muffin, I am practically starting over each time. Luckily, each adventure feels slightly more beatable (exp points?), since I have a few more examples I can use for myself. Masnevets and I had a good discussion about a few basic examples of adjoint functors (recall the definition here: basically, we need a pair of functors F \colon C \to D and G \colon D \to C such that we have a natural isomorphism \hom_D(X, F(Y)) \cong \hom_C(G(X), Y)), and thus we have a new Concrete Nonsense post.

Before we start, I want to state that I’m trying something new. This post is not intended to be an introduction to adjoints as I originally envisioned – I realized that there are many better sources for that. Instead, I’ll try to do a free association that juxtoposes a few elementary concepts. You don’t even have to know the definitions of adjoints to start seeing what I’m getting to, since I’ll be namedropping algebraic structures like Kanye West.

My first introduction to adjoints was from algebraic topology, where you naturally bump into the functors \otimes_R and \hom(R, -). It was unnecessary at the time (for the scope of the course, at least) to know that they were adjoints, but now I know them as the “tensor-hom adjunction pair” (saying “tensor-hom” a lot helps me remember tensor as the left adjoint and hom as the right). Furthermore,  knowing this relationship allows me to remember some other things – in particular, knowing the left- and right- exactness of these functors, which I used to always mix up. Left adjoints are always right-exact, and right adjoints are always left-exact. Combined with knowing that tensoring is a left-adjoint, I now know that tensoring is right-exact and adjoints are left-exact.

I’ll now make a digression pertaining to something which I believe is important but seldom discussed. By now, some readers are probably complaining that my previous statement doesn’t make it easier to remember anything, because I have to remember an equally arbitrary fact – and what if our brains find it more intuitive that “left adjoints would be left-exact,” which is wrong? In my experiences, the sets of things that are easy to remember for mathematicians are extremely different from one person to the next. Thus, it may help (especially for “elementary” topics such as this one) to just chain lots of little ideas together, so if someone links concepts A, B, and C, and Alice finds it easier to remember B and C from A, she’ll benefit just as much as Bob, who likes to remember A and C from B (and Chris is sad that nobody likes C except him). As math presentation tends to be fairly “structured” – no surprise, knowing mathematicians – I wouldn’t mind seeing more “free associations” in print or the blogosphere, because this is the form of communication which reminds me most of chatting about math with other people while drinking coffee in the common room, an experience which has often taught me a lot more than going to most lectures.

(since this is a free association, before we go back to adjoints, we might as well see another way to remember the exactness assignments from Masnevets. A good “toy exact sequence” to use here is 0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0, where the second map is multiplication by 2 and the third is taking mods . Tensoring by \mathbb{Z}/2\mathbb{Z} turns the second map into the zero map, so tensoring is not left-exact and thus must be right-exact.)

Many of you may be surprised by tensor-hom as my first choice of adjoint functors – that’s only because I learned that one first (this is a lie, and we’ll come back to later). There’s a much easier example using the forgetful functor. Consider the forgetful functor G \colon \mathrm{Grp} \to \mathrm{Set}. It happens to be the right adjoint to the functor F \colon \mathrm{Set} \to \mathrm{Grp} that sends a set to a free group generated by the set. The intuition here is this: by the definition of adjunction, we want “the number of maps out of the image of F” to be “the same” as “the number of maps into the image of G.” When G forgets so much structure, we’ll get lots of maps into the set G(Y). To get lots of maps out of F(X), we want a lot of freedom on the structure, and hence the free group. Of course, there’s nothing particularly binding here about groups; we can do this in general to get a “free functor” when we have a forgetful functor from a category of other algebraic structures to sets (we can’t do this *all* the time, but most of the time we are fine. I don’t understand the conditions here too well, which involves an overloading of the word “variety,”  so I won’t expound).

In fact, let’s take it a step further. We don’t only have to forget into sets. We can forget, for example, from abelian groups into groups, or from associative algebras into Lie algebras. What did we forget in these two cases? Commuting relations and the product, respectively. So intuitively, to “match the complexity” of the two \hom‘s, we want to get just as much structure back: we want to make things commute in an arbitrary group; we want to be able to multiply things in a Lie algebra. It was quite a joyful “but of course” when I realized that the left adjoints of the two functors above came out to be abelianization and the universal enveloping algebra, respectively.

Finally, in this forgetful context I’ll return to the actual first example of adjoint functors I’ve seen (though I definitely did not know it in that context at the time). When we restrict a representation Vof G, or equivalently a F[G]-module, to a sub-representation \mathrm{Res} V of H, we’re “forgetting” how the other elements of G act on our vector space. So is there a natural way to get them back? For each representation W of of H we can induce the representation \mathrm{Ind} W. This ends up being another adjunction pair, of course. Here’s an immediate consequence: in the case that these groups are finite, note that the dimension of the two \hom‘s must match; this just gives us \langle \mathrm{Ind W}, V \rangle = \langle W, \mathrm{Res} V \rangle, which is the Frobenius reciprocity formula from second-semester abstract algebra.


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  1. Neat. I was planning on writing a post about Frobenius reciprocity from the adjoint functor perspective, so I’m glad you brought it up. It’s probably the example where the analogy between adjoint functors and adjoint linear transformations is closest, since \text{Res} is a categorified linear transformation taking categorified linear combinations (direct sums) of irreducible representations of G to combinations of irreducible representations of H. This immediately tells you that the adjoint exists – just take the transpose – but I can’t decide whether this is a worse or better way to show that induced representations exist than writing them down without explaining where they come from.

    And while we’re on the subject, does anyone have a good reference for the construction of free Lie algebras?

  2. Qiaochu: Ironically, so was I. But only as an auxiliary thing to preserve self-containedness and to proceed to modular representation theory, so I’ll probably just refer to your post instead.

    I think Serre’s _Lie Algebras and Lie Groups_ is a good reference.

  3. being two of my favorite young expositors, you probably shouldn’t be deterred from writing on the topic if you have something systematic on your queue, because as I said, my post is not anything systematic at all.

    As for free Lie algebras, I feel most coverages do it well. Surprisingly, I’ve heard *bad* things about that particular Serre’s book even though I liked most of his other writing (something about it containing errors and/or being too technical?).

    My *favorite* introduction to Lie algebras is the Springer undergraduate text _Introduction to Lie Algebras_ by Erdmann/Wildon. Amazingly clear, contains great motivation, and sketches many future directions. It even has solved exercises, which I consider to be great from problem-solving-based learners like myself (I absorb very little theory by reading). It is very elementary but not condescending.

  4. Qiaochu: I am not sure I follow your reasoning. Res is a functor which induces a linear transformation from the representation ring of G to the representation ring of H. Hence it has an adjoint in the linear algebra sense, which is some linear transformation from Rep(H) to Rep(G). How do we know a priori that this comes from a functor?

  5. Qiaochu: I also misread your comment and read “universal enveloping algebra” instead of “free Lie algebra.” Fewer sources talk about that… how about Bourbaki? I usually hate their style, but that particular one felt clean (I read this about n years ago so I don’t remember).

  6. I’ll have to take a look at all these suggestions; thanks, guys.

    Steven: well, we don’t. I just meant that this is one way to convince yourself that induced representations should exist.

  7. [...] it is called an adjoint functor. For more about adjoints, see (in no particular order) posts at Concrete Nonsense, the Unapologetic Mathematician, and Topological [...]

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