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Stream: learning: questions

Topic: Representation theory

Paolo Perrone (Sep 23 2024 at 14:00):

Hi all. Does anyone have specific references for usage of categorical ideas in representation theory?

fosco (Sep 23 2024 at 15:02):

I'm not an expert, but I have seen category theory used

• to study representations of the symmetric groups https://arxiv.org/abs/2106.00190
• in connection with (algebraic) topology https://arxiv.org/abs/0801.3843
• in connection with quantum algebra, to let groups act on bicategories https://arxiv.org/abs/1607.05148 and broadly speaking, in "higher geometry" https://arxiv.org/abs/1503.00888

But these are just the papers I know, there are certainly other ideas, connected to this, or not.

fosco (Sep 23 2024 at 15:04):

The "great divide" that I see in using category theory to understand representation theory cuts through the discrete-continuous axis: for obvious reasons, studying representations of finite groups requires ideas and techniques from combinatorics and algebra. On the other side, for equally evident reasons, representation theory of topological groups borrows from differential geometry, algebraic topology...

fosco (Sep 23 2024 at 15:09):

A good starting point might be a cross-list search on arXiv

https://arxiv.org/search/advanced?advanced=1&terms-0-term=math.CT&terms-0-operator=AND&terms-0-field=cross_list_category&terms-1-term=math.RT&terms-1-operator=AND&terms-1-field=cross_list_category&classification-mathematics=y&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=200&order=-announced_date_first

Dylan Braithwaite (Sep 23 2024 at 15:10):

Maybe some of the references in [[categorification in representation theory]] would be useful too

Tobias Fritz (Sep 23 2024 at 15:18):

The concept of fusion category is an abstract generalization of the category of representations of a finite group. These and related structures (in particular in braided variants) describe quasi-particles and their interaction in certain condensed matter systems. I don't understand the full details, but a hom-space like $\mathsf{Hom}(A \otimes B, C)$ is roughly something like the amplitude of an $A$-particle and a $B$-particle 'fusing' to a $C$-particle. Probably Why are topological phases described by modular tensor categories? gives a better explanation.

John Baez (Sep 23 2024 at 16:07):

Paolo Perrone said:

Hi all. Does anyone have specific references for usage of categorical ideas in representation theory?

Yikes, what a huge subject! Well, I guess any subject seems large in proportion to how much time you spend on it. Can you say more about exactly what you want to know?

A lot of important and basic stuff is clearly explained in Anderson and Fuller's Rings and Categories of Representations. The category of representations of a group or algebra can be seen as a special case of the category of modules of a ring. These special cases have special features, but it's good to understand categories of modules of rings in detail. This is not merely because the theorems apply throughout representation theory, but because modules of rings are a great place to learn how a lot of general concepts in category theory get used in representation theory: concepts like projectivity and injectivity, simplicity and semisimplicity, artinianness and noetherianness, and Morita equivalence.

fosco (Sep 23 2024 at 19:41):

I suspect Paolo implicitly meant representations of groups and sees module theory as a different topic (whereas I understand why one borrows ideas from the other).

Chris Grossack (they/them) (Sep 23 2024 at 19:45):

You probably know this, but for people new to the area: it's not just that one borrows from the other! The theory of group representations is subsumed by the theory of algebra representations! Indeed, if $G$ is a group, then the data of a representation of $G$ on a $k$-vector space is the same data as the data of a $kG$-module, where here $kG$ is the [[group algebra]].

fosco (Sep 23 2024 at 19:46):

I agree :smile: bad choice of words

Chris Grossack (they/them) (Sep 23 2024 at 19:47):

In fact, more generally if $\mathcal{C}$ is any category that "looks like a category of representations" then there's an algebra $A$ so that $\mathcal{C}$ is the category of $A$-modules.

In a weak sense, this is the famous [[Freyd-Mitchell embedding theorem]]. In a strong sense, this is [[Tannaka Duality]].

Chris Grossack (they/them) (Sep 23 2024 at 19:48):

No worries! I really was sure you knew, haha.

fosco (Sep 23 2024 at 19:49):

In any case, maybe @Paolo Perrone can help by narrowing down the question a bit, were it only because it's hard to mention just one topic where representation theory uses category theory.

Although relatively technical, there is a huge amount of literature on representing quivers, because of this result of Gabriel who paved the way to "tilting theory"

image.png

Chris Grossack (they/them) (Sep 23 2024 at 19:49):

Regardless, this is good motivation for just learning some module theory (read: the theory of representations of algebras) once and for all. It develops itself quite nicely alongside the theory of nice monoidal categories, and translating back and forth between properties of an algebra and properties of its category of modules is incredibly useful (and incredibly cool!)

Chris Grossack (they/them) (Sep 23 2024 at 19:49):

I wish I had a reference for this, but I've basically just picked it up myself through osmosis over the years... I'm sure somebody has written it down cleanly, though

fosco (Sep 23 2024 at 19:50):

Tilting theory is related to Morita equivalence in a derived sense, which is the relation of "having the same derived categories of representations" for two monoids (broadly intended)

fosco (Sep 23 2024 at 19:52):

The derived category of coherent sheaves over $\mathbb P^n(k)$ has a combinatorial model in this sense, being derived-equivalent to the category of modules(=representations) of a very concrete k-algebra

image.png

fosco (Sep 23 2024 at 19:57):

Chris Grossack (they/them) said:

I wish I had a reference for this, but I've basically just picked it up myself through osmosis over the years... I'm sure somebody has written it down cleanly, though

Module theory is the statement that in order to understand a ring $R$, you often want to study the ways $R$ acts on stuff. Or at least, this was how ring theory was introduced to me back in the days.

Of course "we" know this is a meta-principle in disguise, familiar to a category theorist: some unknown property of $\cal C$ is better understood when regarded in $\widehat{\cal C} = [{\cal C}^o,{\bf Set}]$, the category of "modules" over which $\cal C$ "acts"...

...and probably it's even better to study the bicategory of bimodules $[{\cal C}^o\times{\cal D},{\bf Set}]$ :eyes:

fosco (Sep 23 2024 at 19:58):

(btw just for completeness and since Paolo is looking for references: the screenshots I took come from the "Handbook of tilting theory")

Nathanael Arkor (Sep 23 2024 at 20:00):

fosco said:

...and probably it's even better to study the bicategory of bimodules $[{\cal C}^o\times{\cal D},{\bf Set}]$ :eyes:

Or the (virtual) double category of bimodules :)

Chris Grossack (they/them) (Sep 23 2024 at 20:00):

At the risk of continuing to derail this thread, I'll briefly say that this isn't something special about $\mathbb{P}^n$! Lots and lots of varieties $X$ can be profitably studied by seeing $D^b(\text{Coh}(X))$ as equivalent to $D^b(\text{Rep}(Q))$ for some quiver (possibly with relations).

This lets you turn interesting geometric questions about $X$ into questions about $D^b(\text{Coh}(X))$ and thus into questions about $D^b(\text{Rep}(Q))$, which is where you actually do your computations to get an answer (since quiver representations are very concrete and combinatorial).

You can play a similar game with [[fukaya categories]] if you have a symplectic structure rather than a complex structure, and Haiden-Katzarkov-Kontsevich showed (for surfaces) that $D^b(\text{Fuk}(Y))$ is also equivalent to $D^b(\text{Rep}(Q))$ for a quiver with relations.

The fact that $D^b(\text{Coh}(X))$ and $D^b(\text{Fuk}(Y))$ are both related to quivers is a very hands on way to start studying [[homological mirror symmetry]]... But I should really shut up now, since this is the wrong thread for this discussion, haha

Paolo Perrone (Sep 23 2024 at 20:05):

I mostly meant representation theory of groups, but anything is welcome, really, so go crazy!
I'm asking for a friend (really).

John Baez (Sep 23 2024 at 21:17):

fosco said:

The derived category of coherent sheaves over $\mathbb P^n(k)$ has a combinatorial model in this sense, being derived-equivalent to the category of modules(=representations) of a very concrete k-algebra

What is this very concrete k-algebra? It's very tantalizing being told both by you and the article that "there's this very concrete algebra but we won't tell you what it is". :upside_down:

John Baez (Sep 23 2024 at 21:17):

I feel I should know this and I'm annoyed I don't!

John Onstead (Sep 23 2024 at 21:36):

fosco said:

Module theory is the statement that in order to understand a ring R, you often want to study the ways R acts on stuff. Or at least, this was how ring theory was introduced to me back in the days.

Of course "we" know this is a meta-principle in disguise, familiar to a category theorist: some unknown property of C is better understood when regarded in C=[Co,Set], the category of "modules" over which C "acts"...

This almost sounds a little like the Yoneda Lemma and Perspective!

Paolo Perrone said:

I mostly meant representation theory of groups, but anything is welcome, really, so go crazy!

This probably won't be helpful since this is the "trivial" case of the relationship between category theory and representation theory but I find it so cool I wanted to share it here. First, given a group G, we can find a one object category BG, with only automorphisms, such that G is precisely the automorphism group of that one object. So every element of G corresponds to an automorphism in BG, and the group operation in G corresponds to composition in BG. Then a representation of G is a functor from BG to RVect! This functor assigns every automorphism in BG- and thus element in G- to a matrix on some vector space, which we will think of as the matrix representation of that group element. Unfortunately this approach isn't very generalizable since not every math object can be viewed as a one object category of some form, but I still think it's a pretty cool idea!

John Baez (Sep 23 2024 at 23:20):

It's very cool. It is generalizable to modules of rings, which is the main topic of that book I was recommending. For any ring $R$ there is a one-object category enriched over $\mathsf{Ab}$ (the category of abelian groups), called $B R$. Morphisms of $B R$ are the elements of $R$, with composition being multiplication in $R$. A right module of $R$ is then an Ab-enriched functor

$B R \to \mathsf{Ab}$

A left module of $R$ is an $\mathsf{Ab}$-enriched functor

$B R^{\text{op}} \to \mathsf{Ab}$.

So, the category of left modules of $R$ is the category of $\mathsf{Ab}$-enriched presheaves on $R$.

John Baez (Sep 23 2024 at 23:23):

It's very reasonable and Yoneda-esque to pass from an enriched category to the category of enriched presheaves on that category.

John Baez (Sep 23 2024 at 23:23):

(I may be getting left and right mixed up above, but you can straighten it out.)

David Egolf (Sep 23 2024 at 23:25):

Interesting! In a similar spirit, I wonder if one can develop "internal representation theory", to study things that are one object categories (of some flavour) internal to some category.

John Baez (Sep 23 2024 at 23:31):

Hmm, to talk about a 'one object category' internal to a category, it helps for the latter category to have a terminal object. But that's pretty common.

John Baez (Sep 23 2024 at 23:35):

I've probably studied a few cases of internal representation theory without knowing it. For example a Lie group is a group internal to Diff, the category of smooth manifolds. When people study representations of a Lie group $G$ on (say) finite-dimensional vector spaces, they limit themselves to smooth representations, i.e. smooth functors from

• a one-object category internal to Diff, called $B G$

to

• the category of finite-dimensional vector spaces, which is also, I believe, (equivalent to) a category internal to Diff

so they are in some sense working internal to the category of smooth manifolds. I'm not sure exactly what I mean by that, but it feels right.

Mike Shulman (Sep 23 2024 at 23:40):

I would be more inclined to consider those as categories enriched over Diff.

John Baez (Sep 24 2024 at 00:07):

Yeah, me too, but in my attempt to please David I was trying to get away with treating them as internal, and I think we can (up to equivalence), since any discrete set is a manifold.

John Baez (Sep 24 2024 at 00:08):

There are probably better examples to show off the power of internalization!

John Onstead (Sep 24 2024 at 00:26):

John Baez said:

So, the category of left modules of R is the category of Ab-enriched presheaves on R.

Wow this puts things into a really neat perspective! A while ago I checked the wikipedia page for "Yoneda lemma" and it mentioned how the study of rings by their modules over them is an example of something analogous to what the Yoneda lemma is doing. Now I can see the formalized connection here! Originally, I had completely the wrong idea about why this might be because I was thinking in terms of the Yoneda embedding/perspective that an object is defined up to isomorphism by the morphisms going into and out of it (and thus determined by its slice category C/X or X/C). So I was thinking that the fact you could study a ring by modules over it meant there was some connection between Module(R) and Ring/R that meant that, in some sense, modules had an equivalent "power" to describe R as did its generalized elements. But it looks like this thinking was going off into a dead end and the real way to think about this was in terms of an enriched Yoneda lemma. Maybe someday I'll start a discussion about reconstruction theorems and their relatives the representation theorems and how they work in general, but I got a lot of other stuff to get through before then!

fosco (Sep 24 2024 at 05:54):

John Baez said:

fosco said:

The derived category of coherent sheaves over $\mathbb P^n(k)$ has a combinatorial model in this sense, being derived-equivalent to the category of modules(=representations) of a very concrete k-algebra

What is this very concrete k-algebra? It's very tantalizing being told both by you and the article that "there's this very concrete algebra but we won't tell you what it is". :upside_down:

something on the line of "the path algebra of a finite quiver"

Morgan Rogers (he/him) (Sep 24 2024 at 07:29):

David Egolf said:

Interesting! In a similar spirit, I wonder if one can develop "internal representation theory", to study things that are one object categories (of some flavour) internal to some category.

Sounds like studying actions of internal monoids, which is the kind of thing I've been getting into ;)

fosco (Sep 24 2024 at 07:42):

It's also worth mentioning that if you want to pinpoint what categories admit a notion of "an object acting over another" and "what is the semidirect product of an action of $X$ on $Y$", you get in the ballpark of this seminal paper in categorical algebra http://www.tac.mta.ca/tac/volumes/1998/n2/n2.pdf

Paolo Perrone (Sep 24 2024 at 09:46):

Thanks to everyone, this is all very interesting.

John Baez (Sep 24 2024 at 14:43):

fosco said:

John Baez said:

fosco said:

The derived category of coherent sheaves over $\mathbb P^n(k)$ has a combinatorial model in this sense, being derived-equivalent to the category of modules(=representations) of a very concrete k-algebra

What is this very concrete k-algebra? It's very tantalizing being told both by you and the article that "there's this very concrete algebra but we won't tell you what it is". :upside_down:

something on the line of "the path algebra of a finite quiver"

That's what I was hoping. Then the question is just: which quiver? My guess was $A_n$, the one that looks like this:

$\bullet \to \bullet \to \cdots \to \bullet \to \bullet$

John Baez (Sep 24 2024 at 14:45):

$A_n$ has $n$ dots, but I could be making a fencepost error if I guess that $A_n$ is connected to $\mathbb{P}^n(k)$. Let's see: $A_n$ is the Dynkin diagram of the algebraic group $\mathrm{SL}(n+1,k)$, which acts as symmetries of $\mathbb{P}^n(k)$. So maybe the two fencepost errors cancel here.

John Baez (Sep 24 2024 at 14:47):

Paolo Perrone said:

Thanks to everyone, this is all very interesting.

I asked what you want to learn about representation theory, and I'm still curious about that.

Paolo Perrone (Sep 24 2024 at 16:19):

John Baez said:

Paolo Perrone said:

Thanks to everyone, this is all very interesting.

I asked what you want to learn about representation theory, and I'm still curious about that.

The high-level question is: how can category theory help us in finding invariants of things (group actions, dynamical systems, etc)?

Mike Shulman (Sep 24 2024 at 16:52):

I'm not really following this thread, but I was amused to see this question:

how can category theory help us in finding invariants of things?

at the same time as in another thread we saw

whenever you have a category that's freely generated by something-or-other, any particular something-or-other somewhere else determines an invariant of objects of that category.

Chris Grossack (they/them) (Sep 24 2024 at 17:17):

John Baez said:

fosco said:

The derived category of coherent sheaves over $\mathbb P^n(k)$ has a combinatorial model in this sense, being derived-equivalent to the category of modules(=representations) of a very concrete k-algebra

What is this very concrete k-algebra? It's very tantalizing being told both by you and the article that "there's this very concrete algebra but we won't tell you what it is". :upside_down:

I guess I'll go ahead and put you out of your misery! It turns out that $D^b(\text{Coh}(\mathbb{P}^1)) \simeq D^b(\text{Rep}(\bullet \rightrightarrows \bullet))$, where $\bullet \rightrightarrows \bullet$ is often called the Kronecker Quiver.

More generally, you need to work with "quivers with relations", but this really isn't that much harder! For example, $D^b(\text{Coh}(\mathbb{P}^n))$ is equivalent to derived representations of the Beilinson Quiver, which is harder to typeset in zulip. It has

• vertices labelled $0$ through $n$
• from $i \to i+1$ there are $n+1$-many arrows, labelled $x_1^{(i)}, \ldots, x_{n+1}^{(i)}$
• no other arrows
• relations saying $x_k^{(i+1)} x_j^{(i)} = x_j^{(i+1)} x_k^{(i)}$ for every $i,j,k$.

Chris Grossack (they/them) (Sep 24 2024 at 17:18):

Screenshot-2024-09-24-at-10.18.35-AM.png

Chris Grossack (they/them) (Sep 24 2024 at 17:26):

Where does this come from? The idea is that the line bundles $\mathcal{O}(i)$ for $i = 0, \ldots, n$ generate $\text{Coh}(\mathbb{P}^n)$. So in some sense understanding the whole category reduces to understanding these $\mathcal{O}(i)$ and maps between them really well. But $\mathcal{O}(i)$ is "just" the module of homogeneous degree $i$ polynomials in $n$ variables, and there's a bunch of obvious maps between these: If $p \in \mathcal{O}(i)$ is a degree $i$ homogeneous polynomial, then for any variable $x_j$ the product $x_j p$ is a degree $i+1$ homogeneous polynomial. And of course $x_k x_j p = x_j x_k p$, so if we think of the vertices of our quiver as the $\mathcal{O}(i)$ and the maps between them as "multiply by $x_j$", then we see (roughly) where this quiver (with relations) comes from.

Chris Grossack (they/them) (Sep 24 2024 at 17:32):

You can play this game for dg categories (stable $\infty$-categories, etc) more broadly as well, though to get a small quiver you need to get your hands on a small generating set that you understand well. See this MO post for more:
https://mathoverflow.net/questions/268531/quiver-representations-and-coherent-sheaves

Kevin Carlson (Sep 24 2024 at 17:33):

It's mildly annoying that representation theorists felt the need to make up the name "quiver" for "graph" and "quiver with relations" for "presentation for a $k$-linear category." But so it goes.

Jorge Soto-Andrade (Sep 24 2024 at 17:41):

Hi, I would suggest looking at the work of Andrei Zelevinski (who I knew
well) and Bernstein. Also Nicolas Libedinsky's work on modular
representations. Moreover, I share the viewpoint of Gelfand and
collaborators who were interested in universal methods which work as well
for, say, matrix groups over any local or finite Field. For instance,
Weil representations, Howe duality, Gelfand Models etc

fosco (Sep 24 2024 at 18:02):

Paolo Perrone said:

John Baez said:

Paolo Perrone said:

Thanks to everyone, this is all very interesting.

I asked what you want to learn about representation theory, and I'm still curious about that.

The high-level question is: how can category theory help us in finding invariants of things (group actions, dynamical systems, etc)?

In many ways: a monoid has a category of left actions (=covariant functors $M\to Set$) and each such functor $X : M \to Set$ has a set of fixpoints (="invariants") $X^M = \bigcap_{m\in M}\{x\in X\mid m.x=x\}$, which turns out to have a universal property in $Set^M$. Dually, "coinvariants" $X/M$ arise taking the set of orbits of the action. When you're acting with a ring or a k-algebra fixpoints/invariants and orbits/coinvariants have a co/homological meaning, see page 7 and 8 here (but also what precedes is a very hands-on intro to Hochschild homology).

fosco (Sep 24 2024 at 18:06):

(link didn't make it in the previous message): https://www.homepages.ucl.ac.uk/~ucahbdo/4thYearProject.pdf

Regarding dynamical systems, we have to agree what we mean. There is a divide between discrete and continuous dynamical systems.

A discrete dynamical system is already a rich structure: a representation of the monoid of natural numbers $\mathbb n$, which by freeness of $\mathbb N$ consists of a set $X$ equipped with an endomorphism $f : X\to X$; the action $\mathbb N\times X\to X$ is defined as $(n,x)\mapsto f^n(x)$. This category has plenty of equivalent presentations... for example, it is at the same time the category of endofunctor algebras (or coalgebras, same thing) for the identity functor, the category of Eilenberg-Moore algebras for the monad $\mathbb N\times-$, the category of coalgebras for the comonad $(-)^{\mathbb N}$...

In every Cartesian closed category one can repeat the same construction defining a "natural number object" as an initial object/algebra.

Moreover, if the ambient category is $Cat$ (categories and functors), one obtains a very elegant description, first noticed by Myles Tierney, of the so-called Spanier-Whitehead stabilization. Consider the category $Cat^{{\bf B}\mathbb N}$ of categories equipped with an endofunctor $S : C\to C$ aptly called "suspension". One can restrict to the subcategory $Cat^{{\bf B}\mathbb Z}$ of those categories equipped with an endofunctor which furthermore is an isomorphism of categories (or, less evil, an adjoint equivalence). It turns out that this subcategory $Cat^{{\bf B}\mathbb Z}\hookrightarrow Cat^{{\bf B}\mathbb N}$ arises from pre-composition with the obvious inclusion $\mathbb N \to\mathbb Z$ (which, if you want, has the universal property of embedding the cancellative monoid of natural numbers in its Grothendieck group...).

But then, this functor has a left adjoint, meaning that every category with a suspension endofunctor can be turned into a category with a suspension equivalence; this is a staple idea in homological algebra.

John Baez (Sep 24 2024 at 18:16):

Kevin Carlson said:

It's mildly annoying that representation theorists felt the need to make up the name "quiver" for "graph"

What's really annoying is that graph theorists felt the need to use the name "graph" to mean "simple graph", which is essentially a set equipped with a reflexive symmetric relation. :upside_down:

This forced representation theorists to make up some other name for "graph".

Category theorists seem to sail on unperturbed by the fact that when they say "graph", graph theorists don't understand what they mean by the word.

John Baez (Sep 24 2024 at 18:19):

(By the way, graph theorists actually think "simple graph" means antireflexive symmetric relation, i.e. a symmetric relation $R$ where no element obeys $R(x,x)$. But these are in bijection to reflexive symmetric relations (at least in classical logic - I don't think intuitionistically!) and the latter form a better category: among other things, they allow you to have maps between simple graphs that map two vertices to the same vertex.)