Measuring comodules -- their applications
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.
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