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Poincaré Duality for Koszul Algebras

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Algebra, Geometry and Mathematical Physics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 85))

Abstract

We discuss the consequences of the Poincaré duality, versus AS-Gorenstein property, for Koszul algebras (homogeneous and non homogeneous). For homogeneous Koszul algebras, the Poincaré duality property implies the existence of twisted potentials which characterize the corresponding algebras while in the case of quadratic linear Koszul algebras, the Poincaré duality is needed to get a good generalization of universal enveloping algebras of Lie algebras. In the latter case we describe and discuss the corresponding generalization of Lie algebras. We also give a short review of the notion of Koszulity and of the Koszul duality for \(N\)-homogeneous algebras and for the corresponding nonhomogeneous versions.

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Acknowledgments

It is a pleasure to thank Roland Berger for his kind advices.

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Authors and Affiliations

  1. Laboratoire de Physique Théorique,  UMR 8627 Université Paris XI,  Bâtiment 210, 91405, Orsay Cedex, France

    Michel Dubois-Violette

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  1. Michel Dubois-Violette
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Correspondence to Michel Dubois-Violette .

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Editors and Affiliations

  1. Informatique et Applications, Université of Haute Alsace, Laboratoire de Mathématiques, Mulhouse, France

    Abdenacer Makhlouf

  2. Tallinn University of Technology, Dept. Mathematics, Tallinn, Estonia

    Eugen Paal

  3. Culture and Communication (UKK) Division of Applied Mathematics, Mälardalen University, The School of Education, Västerås, Sweden

    Sergei D. Silvestrov

  4. Dept. Mathematical Sciences, University of Göteborg, Göteborg, Sweden

    Alexander Stolin

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Dubois-Violette, M. (2014). Poincaré Duality for Koszul Algebras. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds) Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol 85. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55361-5_1

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