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PBW deformations of Koszul algebras over a nonsemisimple ring

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Abstract

Let \(B\) be a generalized Koszul algebra over a finite dimensional algebra \(S\). We construct a bimodule Koszul resolution of \(B\) when the projective dimension of \(S_B\) equals two. Using this we prove a Poincaré–Birkhoff–Witt (PBW) type theorem for a deformation of a generalized Koszul algebra. When the projective dimension of \(S_B\) is greater than two, we construct bimodule Koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and Koszul algebras, and then prove the PBW type theorem. The results obtained can be applied to standard Koszul Artin–Schelter Gorenstein algebras in the sense of Minamoto and Mori (Adv Math 226:4061–4095, 2011).

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Acknowledgments

The authors are very grateful to the referee for his/her careful reading of the manuscript, and valuable comments and suggestions, which greatly enhance the readability and the interest of the article. The first author is supported by a grant from NSFC (No. 11171067), the second and the third authors are supported by FWO.

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Authors and Affiliations

  1. Department of Mathematics, Shaoxing College of Arts and Sciences, Shaoxing, 312000, Zhejiang, China

    Ji-Wei He

  2. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium

    Fred Van Oystaeyen

  3. Department WNI, University of Hasselt, Universitaire Campus, 3590, Diepenbeek, Belgium

    Yinhuo Zhang

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  1. Ji-Wei He
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  2. Fred Van Oystaeyen
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  3. Yinhuo Zhang
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Correspondence to Yinhuo Zhang.

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He, JW., Van Oystaeyen, F. & Zhang, Y. PBW deformations of Koszul algebras over a nonsemisimple ring. Math. Z. 279, 185–210 (2015). https://doi.org/10.1007/s00209-014-1362-y

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  • DOI: https://doi.org/10.1007/s00209-014-1362-y

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