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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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    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.2012-07-10

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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/0911.4377 (generators and relations)

http://arxiv.org/abs/1206.2846 (derived Morita equivalence in the $A_{\infty}$ setting)

http://arxiv.org/abs/0905.2621 (exotic derived categories and Koszul duality)