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Do we know the maximal class of Koszul algebras for which any deformation is Koszul?

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  • $\begingroup$ It would be very helpful if you provide some motivation, say what you already know, and what class of deformations you are interested in, as well as what definition of Koszulness you work with (e.g. are your algebras homogeneous, connected, over a field etc.). Without that, this question is a bit too vague and lazy. $\endgroup$
    – Vladimir Dotsenko
    Commented Feb 20, 2013 at 17:50

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If, by deformation, you mean formal one-parameter deformation (like, say, in deformation quantization), then it is already known that the Koszul duality between the symmetric and the exterior algebra is preserved under deformation, and especially under Kontsevich deformation quantization procedure. This is stated and proven in https://arxiv.org/pdf/0908.2299.pdf (see also the two companion papers https://arxiv.org/pdf/0911.4377 and https://arxiv.org/pdf/1002.2561),, which is itself based on very nice ideas of Shoikhet (see https://arxiv.org/pdf/0708.1634 and https://arxiv.org/pdf/0805.0174).

I can elaborate a bit more if you explain more precisely what you had in mind.

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