Journal Article

MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Ragnar-Olaf Buchweitz, Edward L. Green, Nicole Snashall and Øyvind Solberg

in The Quarterly Journal of Mathematics

Volume 59, issue 4, pages 441-454
Published in print December 2008 | ISSN: 0033-5606
Published online January 2008 | e-ISSN: 1464-3847 | DOI: https://dx.doi.org/10.1093/qmath/ham056
MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

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Let Λ = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution 𝔽 of the graded simple modules over Λ is given in [E. L. Green and Ø. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012–1033]. This resolution is shown to have a ‘comultiplicative’ structure in [E. L. Green, G. Hartman, E. N. Marcos and Ø. Solberg, Resolutions over Koszul algebras, Arch. Math. 85 (2005), 118–127.], and this is used to find a minimal projective resolution ℙ of Λ over the enveloping algebra Λe. Using these results, we show that the multiplication in the Hochschild cohomology ring of Λ relative to the resolution ℙ is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of Λ with respect to a canonical basis over k associated to the resolution 𝔽. The natural map from the Hochschild cohomology to the Koszul dual of Λ is shown to be surjective onto the graded centre of the Koszul dual.

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Subjects: Pure Mathematics

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