Measuring comodules -- their applications
type: Item
instance of: scholarly article
title: Measuring comodules -- their applications
zbMATH Open document ID: 0974.16029
DOI: 10.1016/S0393-0440(00)00024-3
author: Marjorie Batchelor
published in: Journal of Geometry and Physics
publication date: 2001-09-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9806148
review text: A coalgebra \(C\) measures an algebra \(A\) to an algebra \(B\) if there is a linear map from \(C\otimes A\) to \(B\) such that \(c\cdot(aa')=\sum(c_1\cdot a)(c_2\cdot a')\) for \(c\) in \(C\), \(a,a'\) in \(A\), and \(c\cdot 1_A=\varepsilon(c)1_B\), where \(\Delta c=\sum c_1\otimes c_2\) and \(\varepsilon\) is the counit of \(C\). There is a simple extension to the situation where \(D\) is a left \(C\)-comodule via \(\rho(d)=d_{-1}\otimes d_0\), \(M\) is a left \(A\)-module and \(N\) is a left \(B\)-module. Then \(D\) measures \(M\) to \(N\) if there is a linear map from \(D\otimes M\) to \(N\) such that \(d\cdot(am)=\sum(d_{-1}\cdot a)(d_0\cdot m)\) for \(d\) in \(D\), \(a\) in \(A\), \(m\) in \(M\). Categorical descriptions and examples are given. Applications are described to connections on bundles, curvature and loop algebras. Finally she considers dual coalgebras \(A^0\) and dual comodules \(M^0\) of an algebra \(A\) and a left \(A\)-module \(M\) respectively, consisting of linear functions vanishing on a cofinite ideal (respectively on a cofinite submodule). This provides a method of dualizing representations, which when applied to loop algebras yields positive energy representations, and when applied to representations of totally disconnected groups leads to the smooth dual.
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 1563132
zbMATH Keywords: measuring comodules
zbMATH Keywords: left comodules
zbMATH Keywords: dual comodules
zbMATH Keywords: loop algebras
zbMATH Keywords: dual coalgebras
reviewed by: Earl J. Taft
MaRDI profile type: Publication
arXiv ID: math/9806148
cites work: Q5749936
cites work: Q4010907
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description: scientific article