Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang Koszul duality patterns in representation theory. (English) Zbl 0864.17006 J. Am. Math. Soc. 9, No. 2, 473-527 (1996). This paper has three parts. In the first part the authors give a general treatment of Koszul rings and Koszul duality. In the words of the authors a Koszul ring is a \(\mathbb{Z}\)-graded ring which is “as close to being semisimple as it possibly can be”. Among other things they prove: (1) a Koszul ring is quadratic; (2) a quadratic ring \(A\) is Koszul if its Koszul complex is a resolution of \(A_0\); (3) the quadratic dual of a Koszul ring is Koszul; (4) a numerical criteria for a graded ring over a field to be Koszul; and (5) an equivalence between triangulated categories of certain derived categories of \(A\)-modules (resp. \(A^!\)-modules).The second part contains an application of the theory developed in part one to the representation theory of semisimple Lie algebras. They show that if \(\mathcal B\) is the trivial block (or in fact more generally the block corresponding to any fixed central character) in the BGG category \(\mathcal O\) then \(E = \text{Ext}^\bullet_{\mathcal O}(L,L)\) is a Koszul ring. Here \(L\) is the direct sum of all simple modules in \(\mathcal B\)). Moreover, the dual of \(E\) plays the same role in a certain subcategory of \(\mathcal O\). This generalizes the selfduality theorem conjectured by the first two authors (in an unpublished preprint from 1986) and proved by the third author (in a likewise unpublished preprint from 1989). (The authors point out in the introduction that the present paper is the result of their efforts to join these two earlier “partially wrong and quite incomplete” preprints). As an easy consequence of these results the authors prove that Verma modules (as well as other important classes of modules) in \(\mathcal O\) are rigid, i.e. have unique Loewy series [cf. R. S. Irving’s results in Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, 47-65 (1988; Zbl 0706.17006)]. In the third part the authors consider Koszul rings arising from what they call “mixed geometry”. They study here certain categories of mixed perverse sheaves on a variety stratified by affine spaces (such as the flag varieties). Reviewer: H.H.Andersen (Aarhus) Cited in 23 ReviewsCited in 407 Documents MathOverflow Questions: Augmented algebras over semisimple ring MSC: 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B35 Universal enveloping (super)algebras 14M15 Grassmannians, Schubert varieties, flag manifolds 22E46 Semisimple Lie groups and their representations Keywords:Koszul rings; Koszul duality; triangulated categories; representation theory of semisimple Lie algebras; BGG category; selfduality theorem; Verma modules; Loewy series; categories of mixed perverse sheaves; flag varieties Citations:Zbl 0706.17006 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Jeffrey Adams, Dan Barbasch, and David A. 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