Koszul duality for extension algebras of standard modules. (English) Zbl 1193.16022
Summary: We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category \(\mathcal O\), and some quasi-hereditary algebras with Cartan decomposition in the sense of König.
MSC:
| 16S37 | Quadratic and Koszul algebras |
| 16E35 | Derived categories and associative algebras |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16W50 | Graded rings and modules (associative rings and algebras) |
Keywords:
finite-dimensional Koszul algebras; global dimension; natural equivalences; bounded derived categories of graded modules; Yoneda extension algebras; Koszul duality; quasi-hereditary algebras; costandard modules; standard modules; homologically orthogonal modules; category \(\mathcal O\)References:
| [1] | Ágoston, I.; Dlab, V.; Lukács, E., Quasi-hereditary extension algebras, Algebr. Represent. Theory, 6, 1, 97-117 (2003) · Zbl 1053.16007 |
| [2] | Beilinson, A.; Ginzburg, V.; Soergel, W., Koszul duality patterns in representation theory, J. Amer. Math. Soc., 9, 2, 473-527 (1996) · Zbl 0864.17006 |
| [3] | Bernsten, I.; Gelfand, I.; Gelfand, S., A certain category of \(g\)-modules, Funktscional. Anal. i Prilozhen., 10, 2, 1-8 (1976), (in Russian) |
| [4] | Boe, B.; Collingwood, D., Multiplicity free categories of highest weight representations. I, II, Comm. Algebra, 18, 4, 947-1032 (1990), 1033-1070 · Zbl 0757.17005 |
| [5] | Cline, E.; Parshall, B.; Scott, L., Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math., 391, 85-99 (1988) · Zbl 0657.18005 |
| [6] | Dixmier, J., Enveloping algebras, (Graduate Studies in Mathematics, vol. 11 (1996), American Mathematical Society: American Mathematical Society Providence, RI), Revised reprint of the 1977 translation · Zbl 0867.17001 |
| [7] | Dlab, V.; Ringel, C. M., Quasi-hereditary algebras, Illinois J. Math., 33, 2, 280-291 (1989) · Zbl 0666.16014 |
| [8] | Dlab, V.; Ringel, C. M., The module theoretical approach to quasi-hereditary algebras, (Representations of Algebras and Related Topics (Kyoto, 1990). Representations of Algebras and Related Topics (Kyoto, 1990), London Math. Soc. Lecture Note Ser., vol. 168 (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 200-224 · Zbl 0793.16006 |
| [9] | Keller, B., Deriving DG categories, Ann. Sci. École Norm. Sup., 27, 1, 63-102 (1994) · Zbl 0799.18007 |
| [10] | König, S., Exact Borel subalgebras of quasi-hereditary algebras. I, Math. Z., 220, 3, 399-426 (1995), with an appendix by Leonard Scott · Zbl 0841.16013 |
| [11] | Kovilyanskaya, H.; Mazorchuk, V., On incidence algebras associated with regular cell decomposition of \(S^n\), Publ. Math. Debrecen, 54, 3-4, 391-402 (1999) · Zbl 0930.16005 |
| [12] | Mazorchuk, V.; Ovsienko, S., A pairing in homology and the category of linear tilting complexes for a quasi-hereditary algebra, J. Math. Kyoto Univ., 45, 711-743 (2005), With an appendix written by C. Stroppel · Zbl 1147.16010 |
| [13] | Reiten, I.; Riedtmann, Ch., Skew group algebras in the representation theory of Artin algebras, J. Algebra, 92, 224-282 (1985) · Zbl 0549.16017 |
| [14] | Rickard, J., Morita theory for derived categories, J. London Math. Soc., 39, 436-456 (1989) · Zbl 0642.16034 |
| [15] | Ringel, C. M., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z., 208, 2, 209-223 (1991) · Zbl 0725.16011 |
| [16] | Rocha-Caridi, A., Splitting criteria for \(g\)-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible \(g\)-module, Trans. Amer. Math. Soc., 262, 2, 335-366 (1980) · Zbl 0449.17008 |
| [17] | Soergel, W., Kategorie \(O\), perverse Garben und Modulnüber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc., 3, 2, 421-445 (1990), (in German) · Zbl 0747.17008 |
| [18] | Soergel, W., Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory, 2, 432-448 (1998) · Zbl 0964.17018 |
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