Recent zbMATH articles in MSC 58Bhttps://zbmath.org/atom/cc/58B2024-03-13T18:33:02.981707ZWerkzeugOn the van Est analogy in Hopf cyclic cohomologyhttps://zbmath.org/1528.160062024-03-13T18:33:02.981707Z"Moscovici, Henri"https://zbmath.org/authors/?q=ai:moscovici.henriThis paper presents results which are analogous to van Est isomorphism between Lie algebra cohomology, but for Hopf cyclic cohomology. First, the author recalls the relevant results about the Hopf algebra \(H_n\) and Hopf cyclic cohomology. Then he explains the role of the van Est isomorphism. He also proves the main results relating the Hopf cyclic cohomology of the Hopf algebra \(H_n\) of moving frames with that of the DG Hopf algebra of forms in \(\mathrm{GL}(n,R)\). Finally, the author illustrates these results with examples of representative cocycles.
For the entire collection see [Zbl 1507.19001].
Reviewer: Angela Gammella-Mathieu (Metz)Riemannian embeddings in codimension one as unbounded \(KK\)-cycleshttps://zbmath.org/1528.580062024-03-13T18:33:02.981707Z"van Suijlekom, Walter D."https://zbmath.org/authors/?q=ai:van-suijlekom.walter-daniel"Verhoeven, Luuk S."https://zbmath.org/authors/?q=ai:verhoeven.luuk-sSummary: Given a codimension one Riemannian embedding of Riemannian spin\(^c\)-manifolds \(\imath:X \to Y\) we construct a family \(\{\imath_!^\varepsilon\}_{0< \varepsilon<\varepsilon_0}\) of unbounded \(K\!K\)-cycles from \(C(X)\) to \(C_0(Y)\), each equipped with a connection \(\nabla^\varepsilon\) and each representing the shriek class \(\imath_!\in K\!K(C(X), C_0(Y))\). We compute the unbounded product of \(\imath_!^\varepsilon\) with the Dirac operator \(D_Y\) on \(Y\) and show that this represents the \(K\!K\)-theoretic factorization of the fundamental class \([X]=\imath_! \otimes [Y]\) for all \(\varepsilon\). In the limit \(\varepsilon\to 0\) the product operator admits an asymptotic expansion of the form \(\frac{1}{\varepsilon}T+D_X+\mathcal{O}(\varepsilon)\) where the ``divergent'' part \(T\) is an index cycle representing the unit in \(K\!K(\mathbb{C},\mathbb{C})\) and the constant ``renormalized'' term is the Dirac operator \(D_X\) on \(X\). The curvature of \((\imath_!^\varepsilon,\nabla^\varepsilon)\) is further shown to converge to the square of the mean curvature of \(\imath\) as \(\varepsilon\to 0\).