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On measurings of algebras over operads and homology theories

Abhishek Banerjee and Surjeet Kour

Algebraic & Geometric Topology 22 (2022) 1113–1158

The notion of a measuring coalgebra, introduced by Sweedler, induces generalized maps between algebras. We begin by studying maps on Hochschild homology induced by measuring coalgebras. We then develop a notion of measuring coalgebra between Lie algebras and use it to obtain maps on Lie algebra homology. Further, these measurings between Lie algebras satisfy nice adjoint-like properties with respect to universal enveloping algebras.

More generally, we develop the notion of measuring coalgebras for algebras over any operad 𝒪. When 𝒪 is a binary and quadratic operad, we show that a measuring of 𝒪–algebras leads to maps on operadic homology. In general, for any operad 𝒪 in vector spaces over a field K, we construct universal measuring coalgebras to show that the category of 𝒪–algebras is enriched over K–coalgebras. We develop measuring comodules and universal measuring comodules for this theory. We also relate these to measurings of the universal enveloping algebra U𝒪(𝒜) of an 𝒪–algebra 𝒜 and the modules over it. Finally, we describe the Sweedler product C 𝒜 of a coalgebra C and an 𝒪–algebra 𝒜. The object C 𝒜 is universal among 𝒪–algebras that arise as targets of C–measurings starting from 𝒜.

measuring coalgebras, algebras over an operad
Mathematical Subject Classification 2010
Primary: 16T15, 18D50
Received: 12 January 2020
Revised: 1 March 2021
Accepted: 29 March 2021
Published: 25 August 2022
Abhishek Banerjee
Department of Mathematics
Indian Institute of Science
Surjeet Kour
Department of Mathematics
Indian Institute of Technology, Delhi