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Hopf cyclic cohomology for noncompact \(G\)-manifolds with boundary. (English) Zbl 1453.58001

Let \(G\) be a Lie group acting properly on an oriented manifold \(M\) with boundary \(\partial M\). A \(k\)-form \(\omega\) is called a “Dirichlet form” if \(i^{\ast}\omega=0\), where \(i:\partial M \to M\) is the inclusion map. Using Dirichlet forms and the \(G\)-action on \(M\), one can define a Hopf algebroid \(\mathcal{H}(G,M)\).
When \(M\) is without boundary, it was shown by X. Tang et al. [J. Noncommut. Geom. 7, No. 3, 885–905 (2013; Zbl 1292.58003)] that the Hopf cyclic cohomology groups of \(\mathcal{H}(G,M)\) are isomorphic to the \(G\)-invariant de Rham cohomology of \(M\). In the article under review, the author proves that, with the presence of the boundary, the Hopf cyclic cohomology groups of this Hopf algebroid are isomorphic to the de Rham cohomology of \(G\)-invariant Dirichlet forms on \(M\) (Proposition 2.2). Using the boundaryless-double of \(M\), the author deduces that these cyclic cohomology groups are of finite dimension, provided the action is cocompact as well (Theorem 1.1).

MSC:

58A12 de Rham theory in global analysis
58A14 Hodge theory in global analysis

Citations:

Zbl 1292.58003
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References:

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