What is it about?
In this paper, we define a calculus, called Koszul calculus, on quadratic algebras, generalizing the Hochschild-Gerstenhaber (HG) calculus of Koszul algebras. The HG calculus is a particular case of Tamarkin-Tsygan calculus. We give an example of non-Koszul quadratic algebra for which the Koszul calculus is different to the HG calculus.
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Why is it important?
The main difference lies on the non-acyclicity of the higher version of Koszul calculus, while acyclicity always holds in the higher HG calculus. So, Koszul calculus provide new invariants. As an application of Koszul calculus, we show that a well-known duality theorem between the HG calculus of a Koszul algebra and its dual, holds for any quadratic algebra endowed with the Koszul calculus. In other words, the true nature of this duality theorem does not depend on any assumption on quadratic algebras.
Perspectives
My current research concerns on extension of Koszul calculus ot N-homogeneous algebras, i.e., on graded algebras generated in degree 1, with homogeneous relations of degree N. A first version of this extension is available on arXiv:1610.01035.
Roland Berger
Universite de Saint-Etienne
Read the Original
This page is a summary of: KOSZUL CALCULUS, Glasgow Mathematical Journal, October 2017, Cambridge University Press,
DOI: 10.1017/s0017089517000167.
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