Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology

Hermann, Reiner

doi:10.1090/memo/1151

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Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology

About this Title

Reiner Hermann, Institutt for matematiske fag, NTNU, 7491 Trondheim, Norway

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 243, Number 1151
ISBNs: 978-1-4704-1995-0 (print); 978-1-4704-3450-2 (online)
DOI: https://doi.org/10.1090/memo/1151
Published electronically: May 13, 2016
Keywords: Exact categories; Gerstenhaber algebras; Hochschild cohomology; Homological algebra; Hopf algebras; Monoidal categories.
MSC: Primary 16E40; Secondary 14F35, 16T05, 18D10, 18E10, 18G15.

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Abstract

In this monograph, we extend S. Schwede’s exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm {Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid.

As a main result, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories. We use our main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasi-triangular bialgebras, we further determine a significant part of the Lie bracket’s kernel, and thereby prove a conjecture by L. Menichi. Along the way, we introduce $n$-extension closed and entirely extension closed subcategories of abelian categories, and study some of their properties.

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References

\bibcomment The bibliography has been divided into two parts, namely the main references, which are explicitly pointed to within the monograph, and the supplemental references, which are for further reading. The supplemental references extend the main pool by additional references concerning ring theoretical and homological methods that are relevant for the study of (the Lie bracket in) Hochschild cohomology, as well as by interpretations and occurrences of the bracket in various fields of mathematics. It is the author’s strong believe that the articles below, and the references therein, lead to, if not cover, a major part of the important material on the subject that has been discussed in the present monograph. \endbibcomment \bibcomment\section*Main references

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Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology

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Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology