Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T01:53:07.711Z Has data issue: false hasContentIssue false
  • English
  • Français

HIGHER KOSZUL DUALITY FOR ASSOCIATIVE ALGEBRAS

Published online by Cambridge University Press:  01 October 2013

VLADIMIR DOTSENKO  and
BRUNO VALLETTE
VLADIMIR DOTSENKO
Affiliation:
School of Mathematics, Trinity College Dublin, Dublin 2, Ireland e-mail: [email protected]
BRUNO VALLETTE
Affiliation:
Laboratoire J. A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Primary 18G10Secondary 13P1018D5018G55
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue A: New developments in Noncommutative Algebra and its applications. A conference in honour of Kenny Brown's and Toby Stafford's 60th birthdays. , October 2013 , pp. 55 - 74
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Beilinson, A., Ginsburg, V. and Schechtman, V., Koszul duality, J. Geom. Phys. 5 (3) (1988), 317–350.Google Scholar
2.Berger, R., Koszulity for non-quadratic algebras, J. Algebra 239 (2) (2001), 705–734.CrossRefGoogle Scholar
3.Berger, R., Dubois-Violette, M. and Wambst, M., Homogeneous algebras, J. Algebra 261 (1) (2003), 172–185.Google Scholar
4.Berger, C. and Moerdijk, I., On the derived category of an algebra over an operad, Georgian Math. J. 16 (1) (2009), 13–28.Google Scholar
5.Bergman, G., The diamond lemma for ring theory, Adv. Math. 29 (2) (1978), 178–218.Google Scholar
6.Bokut, L. A., Imbeddings into simple associative algebras, Algebra i Logika 15 (1976), 117–142.Google Scholar
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), 471–511.Google Scholar
8.Conner, A. and Goetz, P., A-infinity algebra structures associated to $\mathcalK_2$ algebras, J. Algebra 337 (1) (2011), 63–81.Google Scholar
9.Dotsenko, V. and Khoroshkin, A., Gröbner bases for operads, Duke Math. J. 153 (2) (2010), 363–396.Google Scholar
10.Gerritzen, L., Tree polynomials and non-associative Gröbner bases, J. Symb. Comp. 41 (2006), 297–316.Google Scholar
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), 439–449.Google Scholar
12.He, J. W. and Lu, D. M., Higher Koszul algebras and A-infinity algebras, J. Algebra 293 (2) (2005), 335–362.Google Scholar
13.Hoffbeck, E., A Poincaré–Birkhoff–Witt criterion for Koszul operads, Manuscripta Math. 131 (1–2) (2010), 87–110.Google Scholar
14.Husemoller, D., Moore, J. C. and Stasheff, J., Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113–185.Google Scholar
15.Keller, B., Introduction to A-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), 1–35.CrossRefGoogle Scholar
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.Google Scholar
17.Koszul, J.-L., Homologie et cohomologie des algèbres de Lie, Bull. de la Société Mathématique de France 78 (1950), 65–127.Google Scholar
18.Loday, J.-L. and Vallette, B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften, vol. 346 (Springer-Verlag, Berlin, Germany, 2012).Google Scholar
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), 91–128.Google Scholar
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), 2017–2037.Google Scholar
21.Lü, J. F., He, J. W. and Lu, D. M., Piecewise–Koszul algebras, Sci. China Ser. A Math. 50 (12) (2007), 1795–1804.Google Scholar
22.Priddy, S. B., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60.Google Scholar
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.)Google Scholar
24.Quillen, D. G., Homotopical algebra, Lecture Notes in Mathematics, No. 43 (Springer-Verlag, Berlin, Germany, 1967).Google Scholar
25.Rey, A. and Solotar, A., (a,b)-Koszul algebras (Preprint) arXiv:1007.3426.Google Scholar
26.Tate, J., Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–27.Google Scholar
27.Ye, Y. and Zhang, P., Higher Koszul complexes, Sci. China Ser. A 46 (1) (2003), 118–128.Google Scholar