One of the famous theorems that tend to crop up in undergraduate algebra classes is the Artin-Wedderburn theorem, which says
Theorem. Any semi-simple ring is a product of matrix algebras over division algebras. In particular, if is an algebraically closed field, any semi-simple -algebra is a product of matrix algebras over .
(We say that an algebra is semi-simple if any submodule of any -module has a complement, that is, if every short exact sequence of -modules splits).
Now, looking at this theorem, one might imagine that we now know a lot about finite-dimensional algebra. After all, there are only two kinds of finite-dimensional algebras, semi-simple and non-semi-simple, and we understand one of those halves quite well. Better yet semi-simplicity is an “open” condition. If we think about the set of associative products a finite dimensional vector space could have, the set of such products which are semi-simple is an open set in the Zariski topology, which those of us who like algebraic geometry know means it is pretty darn big, provided it is non-empty (which is it is, since every vector space has a semi-simple product as the sum of a bunch of copies of the field).
But, of course, this is ridiculous. To borrow a metaphor, dividing algebras into semi-simple and not-semi-simple is like dividing the world into bananas and non-bananas. Each finite-dimensional algebra has a unique semi-simple quotient, dividing out by the Jacobson radical (the ideal of elements which act trivially on all simple representations), but the number of different ways of attaching a Jacobson radical to a semi-simple representation is totally intractable as a classification problem.
Interestingly, this sort of phenomenon shows up in several places in the world of algebra: any connected linear algebraic group is an extension of a semi-simple group (which we have a pretty good handle on) and a unipotent group (which it is completely hopeless to try to classify) and any finite group is an extension of an almost semi-simple finite group (which are about as hard to understand as the classification of finite simple groups, which is pretty hard, but not entirely hopeless) by a solvable group (which you can pretty conclusively forget about classifying right now).
Now, just because the general classification is hopeless doesn’t mean that we should give up entirely on finite dimensional algebras. We just have to reduce our expectations a bit, and specialize. For reasons that I don’t have time to get to in this post, I’d like to restrict to the case where is graded by non-negative integers, with semi-simple, and generated over by .
Now, there’s no way for such an algebra to be interesting and semi-simple. In fact, the ideal of positively graded elements is the Jacobson radical of .
Instead, we can identify a class of these algebras which “as semi-simple as possible” called Koszul algebras (the correct pronunciation of “Koszul” is a hotly debated topic, made more difficult by the fact that the mathematician in question was French, though this is clearly not a French name. It seems to be some kind of variant of the Polish word for “shirt,” and my limited knowledge of Polish pronunciation suggests it should be pronounced “KOSHool,” though “kohZOOL” seems to be more common amongst English speakers. Oh well, I’m a little anal like that. I also insist on pronouncing “Noether” as “nö-tehr,” not “nuther”).
There are a surprising number of different ways to think about the condition of Koszulity. To me, the most natural is to think about is Ext’s between simples. If are simple -modules, then we can consider . This module inherits a grading from the fact that we can take a homogeneous free resolution of , and calculate Ext using this.
Definition. We call Koszul if is concentrated in degree in this grading. That is, if is the direct sum of all simples, then the two natural gradings on the algebra coincide.
We call this algebra the Koszul dual of . Each graded -module has a corresponding dual module . Call limpid (I absolutely refuse to overload “clean” or “pure” any further) if the two natural gradings on this module coincide.
Now, what is nice about Koszul algebras (or more generally, Koszul categories, the representation categories of Koszul algebras)? Well, first of all, they appear in a lot of natural places. For example, virtually every flavor of category O you can think of is Koszul, and there are some very interesting dualities between them (for eample, a regular block of category O is self-dual) as is discussed in a paper of Beilinson, Ginzburg and Soergel called “Koszul duality patterns in representation theory” which also serves as a good introduction to Koszul algebras in general. They also appear in topology (for example, in the work of Goresky, MacPherson and Kottwitz on equivariant cohomology) and combinatorics (Vic Reiner gives a nice talk about this here, featuring some exciting confusion about pronunciation). Much of the BGS paper mentioned above is dedicated to showing that algebras associated to certain sorts of geometry (perverse sheaves on nice sorts of spaces) are always Koszul (this explains the case of category O), and some of my most recent research has been related to Koszul rings showing up in a somewhat related, but also differently flavored context.
But, of course, that’s not enough to really make the notion interesting unless we can say something about the structure of Koszul rings, and about their categories of representations.
Of course, what’s particularly nice about semi-simple algebras is that it’s really easy to calculate between modules; it’s zero unless . For Koszul algebras, things aren’t so lucky, but Ext’s are still controlled by a “nice” algebra.
Theorem. If are graded limpid -modules, then there is an isomorphism preversing the grading arising from the grading on the modules, and sending the homological grading to the sum of the former and the latter. In particular, with the natural grading corresponds to the “diagonal” subalgebra of .
Underlying this is an equivalence of derived categories, that lets us give a similar description for all modules. So, we can always turn Ext computations one side to ones that may be easier on the other (sometimes we can turn them into computations of just plain old maps).
For those of you who like -algebras (by which I mean Mikael Johansson), this equivalence of categories has a very nice interpretation.
Theorem. The structure on is formal.
Proof. All the maps of the structure have to preserve the grading coming from the grading on the projective resolution of . But that’s the same as the homological grading, so all higher products are trivial.