Abstract
The deformation cohomology of a tensor category controls deformations of the constraint of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur–Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra. The results make precise an intriguing connection between the combinatorics of partitions and invariants of the exterior of the general linear algebra observed by Kostant.
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Notes
The deformation cohomology of a tensor category is sometimes called its Davydov-Yetter cohomology.
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Acknowledgements
This paper had a rather difficult early life. Started in 2016, when the second author was a master student at Ohio University it was largely complete by the time the second author moved to Northeastern University. Mostly due to the ineffectiveness of the first author in managing his ever increasing load it took three years to do the final polishing. The first author would like to thank Max Planck Institute for Mathematics (Bonn, Germany) for hospitality during the Summer of 2019, crucial for the completion of this paper. The first author is partially supported by the Simons Foundation. The authors are grateful to Pavel Etingof for his questions, which led to an improvement of the original version. Special thanks are to the referee for many useful suggestions, also making the paper better (e.g. Remark 3.13 was suggested to us by the referee).
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