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On measurings of algebras over operads and homology theories

Banerjee, A and Kour, S (2022) On measurings of algebras over operads and homology theories. In: Algebraic and Geometric Topology, 22 (3). pp. 1113-1158.

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Official URL: https://doi.org/10.2140/agt.2022.22.1113


Abstract 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 (Formula presented) and the modules over it. Finally, we describe the Sweedler product (Formula presented) of a coalgebra C and an O–algebra A . The object (Formula presented) is universal among O–algebras that arise as targets of C –measurings starting from A .

Item Type: Journal Article
Publication: Algebraic and Geometric Topology
Publisher: Mathematical Sciences Publishers
Additional Information: The copyright for this article belongs to Mathematical Sciences Publishers.
Department/Centre: Division of Interdisciplinary Sciences > Interdisciplinary Mathematical Sciences
Date Deposited: 05 Oct 2022 06:14
Last Modified: 05 Oct 2022 06:14
URI: https://eprints.iisc.ac.in/id/eprint/77041

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