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HIGHER KOSZUL DUALITY FOR ASSOCIATIVE ALGEBRAS

  • VLADIMIR DOTSENKO (a1) and BRUNO VALLETTE (a2)
Abstract
Abstract

We present a unifying framework for the key concepts and results of higher Koszul duality theory for N-homogeneous algebras: the Koszul complex, the candidate for the space of syzygies and the higher operations on the Yoneda algebra. We give a universal description of the Koszul dual algebra under a new algebraic structure. For that we introduce a general notion: Gröbner bases for algebras over non-symmetric operads.

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1.Beilinson A., Ginsburg V. and Schechtman V., Koszul duality, J. Geom. Phys. 5 (3) (1988), 317350.
2.Berger R., Koszulity for non-quadratic algebras, J. Algebra 239 (2) (2001), 705734.
3.Berger R., Dubois-Violette M. and Wambst M., Homogeneous algebras, J. Algebra 261 (1) (2003), 172185.
4.Berger C. and Moerdijk I., On the derived category of an algebra over an operad, Georgian Math. J. 16 (1) (2009), 1328.
5.Bergman G., The diamond lemma for ring theory, Adv. Math. 29 (2) (1978), 178218.
6.Bokut L. A., Imbeddings into simple associative algebras, Algebra i Logika 15 (1976), 117142.
7.Buchberger B., An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal, PhD Thesis (University of Innsbruck, Austria, 1965) (German); J. Symb. Comput., 41 (2006), 471511.
8.Conner A. and Goetz P., A-infinity algebra structures associated to inline-graphic$\mathcalK_2$ algebras, J. Algebra 337 (1) (2011), 6381.
9.Dotsenko V. and Khoroshkin A., Gröbner bases for operads, Duke Math. J. 153 (2) (2010), 363396.
10.Gerritzen L., Tree polynomials and non-associative Gröbner bases, J. Symb. Comp. 41 (2006), 297316.
11.Green E. L. and Marcos E. L., d-Koszul algebras, 2-d-determined algebras and 2-d-Koszul algebras, J. Pure Appl. Algebra 215 (4) (2011), 439449.
12.He J. W. and Lu D. M., Higher Koszul algebras and A-infinity algebras, J. Algebra 293 (2) (2005), 335362.
13.Hoffbeck E., A Poincaré–Birkhoff–Witt criterion for Koszul operads, Manuscripta Math. 131 (1–2) (2010), 87110.
14.Husemoller D., Moore J. C. and Stasheff J., Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113185.
15.Keller B., Introduction to A-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), 135.
16.Keller B., Koszul duality and coderived categories (after K. Lefèvre) (2003). Available at http://www.math.jussieu.fr/keller/publ/kdc.pdf, accessed 15 January 2012.
17.Koszul J.-L., Homologie et cohomologie des algèbres de Lie, Bull. de la Société Mathématique de France 78 (1950), 65127.
18.Loday J.-L. and Vallette B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften, vol. 346 (Springer-Verlag, Berlin, Germany, 2012).
19.Lu D. M., Palmieri J. H., Q. S. Wu and J. J. Zhang, A-infinity algebras for ring theorists, Proceedings of the International Conference on Algebra, Algebra Colloq. 11 (1) (2004), 91128.
20.Lu D. M., Palmieri J. H., Q. S. Wu and J. J. Zhang, A-infinity structure on Ext-algebras, J. Pure Appl. Algebra 213 (11) (2009), 20172037.
21. J. F., He J. W. and Lu D. M., Piecewise–Koszul algebras, Sci. China Ser. A Math. 50 (12) (2007), 17951804.
22.Priddy S. B., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 3960.
23.Prouté A., A -structures, modèle minimal de Baues-Lemaire et homologie des fibrations, PhD Thesis (Université Denis Diderot, Paris 7, 1986). (Reprinted in Theory Appl. Categ. 21 (2011), 199.)
24.Quillen D. G., Homotopical algebra, Lecture Notes in Mathematics, No. 43 (Springer-Verlag, Berlin, Germany, 1967).
25.Rey A. and Solotar A., (a,b)-Koszul algebras (Preprint) arXiv:1007.3426.
26.Tate J., Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 1427.
27.Ye Y. and Zhang P., Higher Koszul complexes, Sci. China Ser. A 46 (1) (2003), 118128.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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