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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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Table of Contents

Chapters

  • Introduction
  • 1. Prerequisites
  • 2. Extension categories
  • 3. The Retakh isomorphism
  • 4. Hochschild cohomology
  • 5. A bracket for monoidal categories
  • 6. Application I: The kernel of the Gerstenhaber bracket
  • 7. Application II: The $\mathbf {\operatorname {Ext}\nolimits }$-algebra of the identity functor
  • A. Basics

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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