Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Quadratic duals, Koszul dual functors, and applications

Author(s): Volodymyr Mazorchuk; Serge Ovsienko; Catharina Stroppel
Journal: Trans. Amer. Math. Soc. 361 (2009), 1129-1172.
MSC (2000): Primary 16S37, 18E30, 16G20, 17B67
Posted: October 8, 2008
MathSciNet review: 2457393
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: This paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and Koszul duality functors backed up by explicit examples. This generalizes the work of Beilinson, Ginzburg, and Soergel, 1996, in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also define a ``Koszul'' duality functor for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category $ \mathcal{O}$ in a quite elementary and explicit way. From this we deduce a conjecture of Bernstein, Frenkel, and Khovanov, 1999. As applications we propose a definition of a ``Koszul'' dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category $ \mathcal{O}$.

References:

[AS]
H. H. Andersen, C. Stroppel, Twisting functors on $ \mathcal{O}$, Represent. Theory 7 (2003), 681-699 (electronic). MR 2032059 (2004k:17010)

[Au]
M. Auslander, Representation theory of Artin algebras. I, II. Comm. Algebra 1 (1974), 177-268; 1 (1974), 269-310. MR 0349747 (50:2240)

[AR]
M. Auslander, I.Reiten, Stable equivalence of dualizing $ R$-varieties. Advances in Math. 12 (1974), 306-366. MR 0342505 (49:7251)

[Ba1]
E. Backelin, Koszul duality for parabolic and singular category $ \mathcal{O}$, Represent. Theory 3 (1999), 139-152 (electronic). MR 1703324 (2001c:17034)

[Ba2]
E. Backelin, The Hom-spaces between projective functors, Represent. Theory 5 (2001), 267-283 (electronic). MR 1857082 (2002f:17007)

[Bass]
H. Bass, Algebraic $ K$-theory, Benjamin, New York-Amsterdam, 1968. MR 0249491 (40:2736)

[BGS]
A. Beilinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR 1322847 (96k:17010)

[BFK]
J. Bernstein, I. Frenkel, M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $ U(\mathfrak{sl}_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199-241. MR 1714141 (2000i:17009)

[BG]
J. Bernstein, S. Gelfand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), no. 2, 245-285. MR 581584 (82c:17003)

[BGG1]
I. Bernšteĭn, I. Gelfand, S. Gelfand, Algebraic vector bundles on $ {\bf P}\sp{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen., 12, (1978), no. 3, 66-67. MR 509387 (80c:14010a)

[BGG2]
I. Bernšteĭn, I. Gelfand, S. Gelfand, A certain category of $ {\mathfrak{g}}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1-8. MR 0407097 (53:10880)

[BoGa]
K. Bongartz, P. Gabriel, Covering spaces in representation theory. Invent. Math. 65 (1981/82), no. 3, 331-378. MR 643558 (84i:16030)

[Br]
G. Bredon, Equivariant cohomology theories, Lecture Notes in Math, 34, 1967. MR 0206946 (34:6762)

[CM]
C. Cibils, E. Marcos, Skew category, Galois covering and smash product of a $ k$-category, Proc. AMS, 134 (2006), no.1, 39-50. MR 2170541 (2006e:18003)

[De]
P. Deligne, Cohomologie a support propre et construction du foncteur $ f^!$, Lecture Notes in Mathematics 20, 1966, pp. 404-423.

[tDi]
T. tom Dieck, Über projektive Moduln und Endlichkeitshindernisse bei Transformationsgruppen, Manuscripta Mathematica 34 (1981), 135-155. MR 620445 (82k:57028)

[Fi1]
P. Fiebig, Centers and translation functors for the category $ \mathcal{O}$ over Kac-Moody algebras, Math. Z. 243 (2003), no. 4, 689-717. MR 1974579 (2004c:17051)

[Fi2]
P. Fiebig, The combinatorics of category $ \mathcal{O}$ for symmetrizable Kac-Moody algebras, Transf. Groups. 11 (2006), no. 1, 29-49. MR 2205072 (2006k:17040)

[Fl]
G. Fløystad, Koszul duality and equivalences of categories, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2373-2398. MR 2204036 (2006m:16028)

[FKS]
I. Frenkel, M. Khovanov, C. Stroppel, A categorification of finite-dimensional irreducible representations of quantum $ sl(2)$ and their tensor products, Slecta Mathematica (N.S.) 12 (2006), no. 3-4, 379-431. MR 2305608 (2008a:17014)

[GJ]
O. Gabber, A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 261-302. MR 644519 (83e:17009)

[Ga]
P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448. MR 0232821 (38:1144)

[GM]
S. Gelfand, Y. Manin, Methods of homological algebra, Second edition, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. MR 1950475 (2003m:18001)

[GK]
V. Ginzburg, M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203-272. MR 1301191 (96a:18004)

[GMRSZ]
E. Green, R. Martınez-Villa, I. Reiten, Ø. Solberg, D. Zacharia, On modules with linear presentations, J. Algebra 205 (1998), no. 2, 578-604. MR 1632765 (99i:16075)

[GRS]
E. Green, I. Reiten, Ø. Solberg, Dualities on generalized Koszul algebras, Mem. Amer. Math. Soc. 159 (2002), no. 754. MR 1921583 (2004b:16042)

[Ha]
D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, Cambridge, 1988. MR 935124 (89e:16035)

[HI]
J. Herzog, S. Iyengar, Koszul modules. J. Pure Appl. Algebra 201 (2005), no. 1-3, 154-188. MR 2158753 (2006d:13013)

[Ir]
R. Irving, Shuffled Verma modules and principal series modules over complex semisimple Lie algebras, J. London Math. Soc. (2) 48 (1993), no. 2, 263-277. MR 1231714 (94i:17013)

[Ja1]
J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, 750, Springer, Berlin, 1979. MR 552943 (81m:17011)

[Ja2]
J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 3, Springer-Verlag, Berlin, 1983. MR 721170 (86c:17011)

[Jo]
A. Joseph, The Enright functor on the Bernstein-Gelfand-Gelfand category $ \mathcal{O}$, Invent. Math. 67 (1982), no. 3, 423-445. MR 664114 (84j:17005)

[KS]
M. Kashiwara, P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, 292, Springer-Verlag, Berlin, 1994. MR 1299726 (95g:58222)

[Ke1]
B. Keller, On the construction of triangle equivalences, in: Derived equivalences for group rings. Lecture Notes in Math., 1685, 155-176, 1998. MR 1649844

[Ke2]
B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63-102. MR 1258406 (95e:18010)

[Ke3]
B. Keller, Koszul duality and coderived categories (after K. Lefèvre), Preprint 2003.

[KM]
O. Khomenko, V. Mazorchuk, On Arkhipov's and Enright's functors, Math. Z. 249 (2005), no. 2, 357-386. MR 2115448 (2005k:17004)

[McL]
S. Mac Lane, Homology. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. MR 1344215 (96d:18001)

[MVS]
R. Martınez Villa, M. Saorın, Koszul equivalences and dualities. Pacific J. Math. 214 (2004), no. 2, 359-378. MR 2042938 (2005e:16046)

[MVZ]
R. Martınez Villa, D. Zacharia, Approximations with modules having linear resolutions. J. Algebra 266 (2003), no. 2, 671-697. MR 1995131 (2004h:16005)

[Ma]
V. Mazorchuk, Applications of the category of linear complexes of tilting modules associated with the category $ \mathcal{O}$, math.RT/0501220, to appear in Alg. Rep. Theory.

[MO]
V. Mazorchuk and S. Ovsienko, A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra, J. Math. Kyoto Univ. 45 (2005), no. 4, 711-741. MR 2226627 (2007d:16015)

[MS1]
V. Mazorchuk and C. Stroppel, On functors associated to a simple root, J. Algebra 314 (2007), no. 1, 97-128. MR 2331754

[MS2]
V. Mazorchuk and C. Stroppel, Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module, Trans. Amer. Math. Soc. 357 (2005), no. 7, 2939-2973. MR 2139933 (2006g:17012)

[MS3]
V. Mazorchuk and C. Stroppel, A combinatorial approach to functorial quantum $ sl(k)$ knot invariants, arXiv:0709.1971.

[Mi]
B. Mitchell, Rings with several objects. Advances in Math. 8 (1972), 1-161. MR 0294454 (45:3524)

[Ri]
J. Rickard, Morita theory for derived categories. J. London Math. Soc. (2) 39 (1989), no. 3, 436-456. MR 1002456 (91b:18012)

[RH]
S. Ryom-Hansen, Koszul duality of translation and Zuckerman functors. J. Lie Theory 14 (2004), no. 1, 151-163. MR 2040174 (2005g:17018)

[Sc]
H. Schubert, Kategorien. I, Heidelberger Taschenbücher, 66, Springer, Berlin, 1970. MR 0274548 (43:311)

[Sh]
U. Shukla, On the projective cover of a module and related results, Pacific J. Math. 12 1962 709-717. MR 0146235 (26:3757)

[So]
W. Soergel, Kategorie $ \mathcal{O}$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421-445. MR 1029692 (91e:17007)

[St1]
C. Stroppel, Category $ \mathcal{O}$: gradings and translation functors, J. Algebra 268 (2003), no. 1, 301-326. MR 2005290 (2004i:17007)

[St2]
C. Stroppel, Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126 (2005), no. 3, 547-596. MR 2120117 (2005i:17011)

[St3]
C. Stroppel, TQFT with corners and tilting functors in the Kac-Moody case, arXive:math/0605103.

[Su]
J. Sussan, Category $ \mathcal{O}$ and $ sl(k)$ link invariants, arXive:math/0701045.

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16S37, 18E30, 16G20, 17B67

Retrieve articles in all Journals with MSC (2000): 16S37, 18E30, 16G20, 17B67

Additional Information:

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, Box 480, 751 06, Uppsala, Sweden
Email: mazor@math.uu.se

Serge Ovsienko
Affiliation: Department of Mathematics, Kyiv University, 64, Volodymyrska st., 01033, Kyiv, Ukraine
Email: ovsko@voliacable.net

Catharina Stroppel
Affiliation: Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, United Kingdom
Email: cs@maths.gla.ac.uk

DOI: 10.1090/S0002-9947-08-04539-X
PII: S 0002-9947(08)04539-X
Received by editor(s): April 26, 2006
Posted: October 8, 2008
Additional Notes: The first author was partially supported by the Swedish Research Council
The second author was partially supported by the Royal Swedish Academy of Sciences and The Swedish Foundation for International Cooperation in Research and Higher Education (STINT)
The third author was supported by The Engineering and Physical Sciences Research Council (EPSRC)
Copyright of article: Copyright 2008, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia