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Joe Moeller (Feb 24 2021 at 18:43):My friend Ethan Kowalenko and his advisor Carl Mautner just put a new paper on the arxiv. (I know it's not CT exactly, but it has category right in the title)
Category O for Oriented Matroids
We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category O. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden--Licata--Proudfoot--Webster.
John Baez (Feb 24 2021 at 20:31):Neat! It would be nice to see what this category looks like sort of concretely in the simplest nontrivial example.
John Baez (Feb 24 2021 at 20:32):So is it the category of all representations of some finite dimensional algebra?
Joe Moeller (Feb 24 2021 at 20:34):I think usually category O is only certain nice ones.
John Baez (Feb 24 2021 at 20:35):Usually for category O it's certain nice representations of the universal enveloping algebra of a semisimple Lie algebra.
John Baez (Feb 24 2021 at 20:36):But this sentence:
We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category O.
seems to suggest they're looking at all representations of some finite-dimensional algebra.
Either that, or they're writing in a fuzzy style.
John Baez (Feb 24 2021 at 20:37):Since universal enveloping algebras are (almost) never finite-dimensional, it's possible that a finite-dim algebra could have a category of all representations that's analogous to the category of nice representations of some universal enveloping algebra.
Joe Moeller (Feb 24 2021 at 20:56):@Ethan Kowalenko is now here, so he could probably answer.
Ethan Kowalenko (Feb 24 2021 at 21:04):John Baez said:
Neat! It would be nice to see what this category looks like sort of concretely in the simplest nontrivial example.
Hey! glad to answer some questions. For instance, the first nontrivial example of our algebra is actually the finite-dimensional algebra which governs the principle block of BGG category O for sl_2: this is a 5-dimensional algebra, given as the path algebra of a quiver with 2 nodes, say alpha and beta, and an arrow each way between them, subject to the condition that any path passing through beta twice must die
John Baez (Feb 24 2021 at 21:06):Neat. I don't know what the "principal block" is... but that's a cute algebra. Does this algebra have one irreducible representation for each integer (= each point in the weight lattice of ), or something like that?
Ethan Kowalenko (Feb 24 2021 at 21:20):The "principal block" of category O is an example of a "block" of category O: while category O itself is some category of "nice" modules over the universal enveloping algebra of a Lie algebra, category O can be split up into irreducible blocks. (I've not worked in the classical setting much, so I have gaps in my understanding here)
Most of these blocks have one simple object, but for certain parameters (nice central characters of the enveloping algebra) these blocks are the module categories for some finite-dimensional algebras. These algebras are what our algebras are the analogues of
Ethan Kowalenko (Feb 24 2021 at 21:22):So for the algebra I described, there are 2 simple objects, which I think of as "one for each node" but you might also think of it as "one for each element of the Weyl group of sl_2"
Ethan Kowalenko (Feb 24 2021 at 21:24):Again, my experience with actual category O is limited, but I've gotten the feel for it from the introduction from the following paper: https://arxiv.org/abs/1010.2001
John Baez (Feb 24 2021 at 21:29):All I know is a bit about the original category O.
John Baez (Feb 24 2021 at 21:38):The link lists some basic properties of category O for a semisimple algebra, and the definition. What makes some category of representations of something be "like" that category O?
John Baez (Feb 24 2021 at 21:39):Anyway, now I think maybe I get what a "block" of category O is. I didn't know each block was the category of representations of some other algebra!
Ethan Kowalenko (Feb 24 2021 at 21:56):John Baez said:
The link lists some basic properties of category O for a semisimple algebra, and the definition. What makes some category of representations of something be "like" that category O?
I would say "being like category O" should be something like the following: Both hypertoric category O (from the link) and BGG category O (the original) are
:jack-o-lantern: categories of some well-behaved representations of some big algebra
:jack-o-lantern: which can be chopped up into "blocks",
:jack-o-lantern: where the nice blocks are the finite-dimensional module categories over a Koszul algebra,
:jack-o-lantern: and there should be some functors (shuffling and twisting) which tell you that these nice blocks are all equivalent in some way (derived Morita equivalence).
One way to look at our paper might be the following: we are looking at the algebras which might govern the blocks of some sort of category O for oriented matroids, and trying to see if we can prove that they are Koszul and equivalent in the same way
Ethan Kowalenko (Feb 24 2021 at 22:03):John Baez said:
Anyway, now I think maybe I get what a "block" of category O is. I didn't know each block was the category of representations of some other algebra!
Cool, right? and it turns out that the algebras for the nice blocks have centers isomorphic to the cohomology rings of the associated flag variety (in the BGG version) or the associated "hypertoric" variety (in the hypertoric version)
Ethan Kowalenko (Feb 24 2021 at 22:05):For our version, these centers are always the face rings of an underlying (unoriented) matroid
Ethan Kowalenko (Feb 24 2021 at 22:08):Which, now that I've said it, hopefully was more "cool" than it was "jargon"
John Baez (Feb 24 2021 at 22:15):Thanks! In your pumpkin-itemized list, are the "blocks" the same as the "nice blocks", or are only some of the blocks "nice"?
Ethan Kowalenko (Feb 24 2021 at 22:27):Only some of the blocks are nice: in the BGG version, you get a "infinitesimal block" for each central character of the enveloping algebra, which has its number of simple objects at most given by the size of the Weyl group. These infinitesimal blocks may or may not be irreducible, depending on the character.
When I say "nice", i mean that the character is nice (regular integral) so that the infinitesimal block is irreducible. Generically, a character gives a block which can be split into irreducible blocks with one simple each, which is a bit boring (i imagine).