# Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but I would like concrete examples of where Koszul duality is either a nice way of looking at a result, or where it provides us with new results. If someone were to be able to explain for example:

1. Why is the Ext-algebra (Yoneda-algebra) (and the way it characterizes Koszulity) useful?

2. Is there a more elementary way of explaining some of the results and applications in the Beilinson-Ginzburg-Soergel paper Koszul duality patterns in representation theory?

3. I find it hard to grasp Manin's motivations for viewing quadratic algebras as noncommutative spaces, cfr. "Quantum groups and noncommutative geometry". Can anyone comment on this?

4. What are the applications of Koszul algebras? Example: I know that by a result of Fröberg every projective variety has a homogeneous coordinate ring that is Koszul, but what does this do for us really?

Those are some of the examples that spring to mind, but feel free to add other phenomena. Also, let me know if you think there's a way to improve this question.

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see: mathoverflow.net/questions/329/what-is-koszul-duality and Polishchuk and Positselski's book Quadratic Algebras in general, math.stackexchange.com/questions/64064/… for q1. I think this is way too broad, someone could write a long expository article about each of your four questions. Generally speaking, Koszul algebras are special because calculating Ext rings is hard and even writing down tractable projective resolutions is hard whereas once you know something is Koszul, you get them for free. – m_t_ Jul 10 '12 at 10:30
as above comment said, there are way too broad to say why Koszul algebras are nice and a good theory to study. Concerning partiuclar questions: (q1) you can also look into Prof. Keller's "Introduction to A-infinitey algebras and modules", which contains material why we are interested in Ext-algebra of a (dg/A-infinity) module $M$. (For $M = A_0$, and $A$ satisfy certain conditions, then this is the Koszul dual). – Aaron Jul 10 '12 at 14:53
For (q2), IMO, BGS is already quite elementary to most students studying representation theory, if you discard the geometric part of the paper. Their core motivation lies in the relations between category $\mathcal{O}$ of complex semisimple Lie algebra and parabolic analogue of category $\mathcal{O}$, namely they are of derived equivalence. Once we have a derived equivalence, the homological behaviour of the two categories are much easier to understand by virtue of Koszul theory. BGS's work is also important to people who studies Kazhdan-Lusztig polynomial. – Aaron Jul 10 '12 at 14:59
The BGS paper is also one of the most celebrated articles which unite representation theory with the graded setting. And it gives criteria to how one could determine a positively graded algebra is Koszul. Rewritten many of the complication in studying Koszul theory. This has become the first step into generalising Koszul theory to other type of Ext-algebras, by looking into graded structure of algebras, in much more elegant way. – Aaron Jul 10 '12 at 15:02

The Koszul complex and quadratic data arise in deformation quantization of quadratic Poisson structures: you can prove an important result on quantized realization via generators and relations of quadratic data, derived Morita theory of $A_{\infty}$-modules categories or introduce Koszul duality of exotic derived categories of dg structures. If you are interested in these topics you can read
http://arxiv.org/abs/1206.2846 (derived Morita equivalence in the $A_{\infty}$ setting)