Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology

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 both the Lie bracket and the squaring map in Hochschild cohomology are invariants 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.

Modified version of author's PhD thesis (Bielefeld University, December 2013). 159 pages. --- Final version, to appear in "Memoirs of the American Mathematical Society". Corrected a mistake in Section 6 (the main results are not affected) and made minor changes according to the suggestions of the referees

References in corpus (4)

• Lectures on Noncommutative Geometry
• Support varieties and the Hochschild cohomology ring modulo nilpotence
• Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology
• A generalization of Watts's Theorem: Right exact functors on module categories

Cited by in corpus (5)

• Extension Fullness of the Categories of Gelfand-Zeitlin and Whittaker Modules
• Homological epimorphisms, recollements and Hochschild cohomology - with a conjecture by Snashall-Solberg in view
• Some homological properties of category O. III
• Exact sequences, Hochschild cohomology, and the Lie module structure over the $M$-relative center
• Lie brackets on Hopf algebra cohomology