Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology / Reiner Hermann

Auteur: Hermann, Reiner - AuteurType de document: MonographieCollection: Memoirs of the American Mathematical Society ; 1151Langue: anglaisPays: Etats UnisÉditeur: Providence (R.I.) : American Mathematical Society, 2016Description: 1 vol. (V-146 p.) : ill. ; 26 cm ISBN: 9781470419950 ; br. ISSN: 0065-9266Résumé: 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 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.Bibliographie: Bibliogr. p. [141]-146. Index. Sujets MSC: 16E40 Associative rings and algebras -- Homological methods -- (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
16T05 Associative rings and algebras -- Hopf algebras, quantum groups and related topics -- Hopf algebras and their applications
18D10 Category theory; homological algebra -- Categories with structure -- Monoidal categories, symmetric monoidal categories, braided categories
18E10 Category theory; homological algebra -- Abelian categories -- Exact categories, abelian categories
18G15 Category theory; homological algebra -- Homological algebra -- Ext and Tor, generalizations, Künneth formula
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Bibliogr. p. [141]-146. Index

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 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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