Brackets and products from centres in extension categories
Abstract
Building on Retakh's approach to Ext groups through categories of extensions, Schwede reobtained the wellknown Gerstenhaber algebra structure on Ext groups over bimodules of associative algebras both from splicing extensions (leading to the cup product) and from a suitable loop in the categories of extensions (leading to the Lie bracket). We show how Schwede's construction admits a vast generalisation to general monoidal categories with coefficients of the Ext groups taken in (weak) left and right monoidal (or Drinfel'd) centres. In case of the category of left modules over bialgebroids and coefficients given by commuting pairs of braided (co)commutative (co)monoids in these categorical centres, we provide an explicit description of the algebraic structure obtained this way, and a complete proof that this leads to a Gerstenhaber algebra is then obtained from an operadic approach. This, in particular, considerably generalises the classical construction given by Gerstenhaber himself. Conjecturally, the algebraic structure we describe should produce a Gerstenhaber algebra for an arbitrary monoidal category enriched over abelian groups, but even the bilinearity of the cup product and of the Lietype bracket defined by the abstract construction in terms of extension categories remain elusive in this general setting.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.11552
 Bibcode:
 2021arXiv211211552F
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology;
 Mathematics  Quantum Algebra;
 Mathematics  Rings and Algebras
 EPrint:
 42 pages