Recent zbMATH articles in MSC 19https://zbmath.org/atom/cc/192022-11-17T18:59:28.764376ZWerkzeugOn cohomologies and algebraic \(K\)-theory of Lie \(p\)-superalgebrashttps://zbmath.org/1496.170182022-11-17T18:59:28.764376Z"Rakviashvili, Giorgi"https://zbmath.org/authors/?q=ai:rakviashvili.giorgiSummary: An enveloping associative superalgebra \(\Lambda [L,\alpha ,\beta ]\) and its groups of cohomologies are defined and it is proved that there exists Frobenius multiplication of the Quillen algebraic \(K\)-functors of \(\Lambda [L,\alpha ,\beta ]\). These results generalize corresponding results for Lie \(p\)-algebras which were proved by the author earlier.Cyclic Gerstenhaber-Schack cohomologyhttps://zbmath.org/1496.180212022-11-17T18:59:28.764376Z"Fiorenza, Domenico"https://zbmath.org/authors/?q=ai:fiorenza.domenico"Kowalzig, Niels"https://zbmath.org/authors/?q=ai:kowalzig.nielsWe know that deformations of an associative algebra is controlled by its Hochschild cohomology [\textit{M. Gerstenhaber}, Ann. Math. (2) 78, 267--288 (1963; Zbl 0131.27302)]. The same way, deformations of a bialgebra is controlled by Gerstenhaber-Schack cohomology [\textit{M. Gerstenhaber} and \textit{S. D. Schack}, Contemp. Math. 134, 51--92 (1992; Zbl 0788.17009)]. The main problem the authors tackle arises from the question whether Gerstenhaber-Schack cohomology carries a Gerstenhaber bracket analogous to the bracket structure on Hochschild cohomology. The authors show that when the underlying bialgebra is a Hopf algebra the diagonal of the Gerstenhaber-Schack complex carries an operad structure with multiplication (Theorem B). Since bisimplicial objects are homotopic to their diagonals, the Gerstenhaber-Schack cohomology carries a natural Gerstenhaber bracket due to [\textit{M. Gerstenhaber} and \textit{S. D. Schack}, Contemp. Math. 134, 51--92 (1992; Zbl 0788.17009); \textit{J. E. McClure} and \textit{J. H. Smith}, Contemp. Math. 293, 153--193 (2002; Zbl 1009.18009)]. Moreover, they also show that (Theorem C) when the antipode of the underlying Hopf algebra is involutive, or when the Hopf algebra has a modular pair in involution, then operad is cyclic, and therefore, the Gerstenhaber-Schack cohomology supports a Batalin-Vilkovisky algebra structure by [\textit{L. Menichi}, \(K\)-Theory 32, No. 3, 231--251 (2004; Zbl 1101.19003)]. Interestingly, the bracket is trivial when the Hopf algebra is finite dimensional (Theorem D), and therefore, Gerstenhaber-Schack cohomology has a \(e_3\)-algebra structure due to [\textit{D. Fiorenza} and \textit{N. Kowalzig}, Int. Math. Res. Not. 2020, No. 23, 9148--9209 (2020; Zbl 1468.55008)].
Reviewer: Atabey Kaygun (İstanbul)Quasi-representations of groups and two-homologyhttps://zbmath.org/1496.190012022-11-17T18:59:28.764376Z"Dadarlat, Marius"https://zbmath.org/authors/?q=ai:dadarlat.mariusTwo unitaries whose commutator has small norm give a quasi-representation of the group \(\mathbb{Z}^2\). It may happen that these unitaries are not close to exactly commuting unitaries. \textit{R. Exel} and \textit{T. Loring} [Proc. Am. Math. Soc. 106, No. 4, 913--915 (1989; Zbl 0677.15003)] showed that the nonvanishing of a certain winding number obstructs this. The group \(\mathbb{Z}^2\) is the fundamental group of a torus. The author [J. Topol. Anal. 4, No. 3, 297--319 (2012; Zbl 1258.46029)] has extended the Exel-Loring formula to quasi-representations of the fundamental group \(\Gamma_g\) of an oriented surface of genus \(g\ge1\). The first result in this article generalises this result further to a discrete group together with a class \(x\) in its second integral homology. Via the assembly map, \(x\) gives rise to a class in the \(K\)-theory of the \(\ell^1\)-Banach algebra of the group, and a quasi-representation to a finite matrix algebra attaches an integer to this. The first main theorem identifies this integer with the winding number of a certain loop. An analogous theorem holds for quasi-representations into a unital \(C^\ast\)-algebra with a finite trace.
The setup above is linked to surface groups as follows: there are \(g\ge1\) and a group homomorphism \(f\colon \Gamma_g \to \Gamma\) so that the given homology class \(x\) is the \(f_*\)-image of a canonical element in \(H^2(\Gamma_g,\mathbb{Z})\). This is a key idea for the proof.
The second main result in the article shows under some assumptions that there is a quasi-homomorphism for which the resulting integer is not zero. Of course, this only makes sense if \(x\) is not torsion. In addition, it is assumed that the group admits a \(\gamma\)-element and is isomorphic to a subgroup of the unitary group of a quasidiagonal \(C^\ast\)-algebra. Both assumptions hold, for instance, if the group in question is amenable.
Reviewer: Ralf Meyer (Göttingen)The \(\text{KO}\)-valued spectral flow for skew-adjoint Fredholm operatorshttps://zbmath.org/1496.190022022-11-17T18:59:28.764376Z"Bourne, Chris"https://zbmath.org/authors/?q=ai:bourne.chris"Carey, Alan L."https://zbmath.org/authors/?q=ai:carey.alan-l"Lesch, Matthias"https://zbmath.org/authors/?q=ai:lesch.matthias"Rennie, Adam"https://zbmath.org/authors/?q=ai:rennie.adamThe classical spectral flow is defined for a path of self-adjoint Fredholm operators with invertible end points and counts the number of eigenvalues that cross zero along the path. This article studies the spectral flow for paths of Fredholm operators on real Hilbert spaces with Clifford algebra symmetries, taking values in the \(\text{KO}\)-theory of a point instead of the integers. The \(K\)-theory for real \(C^*\)-algebras has received new attention recently because of its application in the study of symmetry-protected topological phases. Certain symmetries of physical systems such as time reversal are implemented by anti-unitary operators on a Hilbert space, and replace complex by real \(C^*\)-algebras. The spectral flow has already been used for such applications in certain situations, which creates the motivation to develop the theory more systematically.
The article is based on classical results describing the \(\text{KO}\)-theory of a point using modules over Clifford algebras and spaces of Fredholm operators that satisfy appropriate commutation relations with respect to a Clifford algebra representation. More precisely, the authors use a pair of complex structures \(J_0\) and~\(J_1\) that anticommute with the Clifford generators and that satisfy \(\|J_0 - J_1\| <2\). They define an index for such a pair and give an alternative formula for it. Then they define the spectral flow for a continuous path of skew-adjoint Fredholm operators with invertible end points, anticommuting with Clifford generators. They compare their definition with previous definitions and treat some examples related to topological phases. They carry the definition over to paths of unbounded operators and carefully treat the continuity of paths of such operators.
Given various approaches to defining the spectral flow, an important observation in the article is that all reasonable definitions of it agree. Here ``reasonable'' means that the definition should be homotopy invariant, additive for concatenation of paths, and be normalised suitably. This allows to identify the spectral flow with other constructions, by proving that these have the relevant properties as well. This idea is used to identify the spectral flow with the index of a certain Fredholm operator built out of the path, generalising a formula by Robbin and Salamon. The spectral flow is also identified with a Kasparov product in bivariant \(K\)-theory.
Reviewer: Ralf Meyer (Göttingen)Induced coactions along a homomorphism of locally compact quantum groupshttps://zbmath.org/1496.460702022-11-17T18:59:28.764376Z"Kitamura, Kan"https://zbmath.org/authors/?q=ai:kitamura.kanSummary: We consider induced coactions on \(C^*\)-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group.Baum-Connes and the Fourier-Mukai transformhttps://zbmath.org/1496.460722022-11-17T18:59:28.764376Z"Emerson, Heath"https://zbmath.org/authors/?q=ai:emerson.heath"Hudson, Daniel"https://zbmath.org/authors/?q=ai:hudson.danielIf \(\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d\) is the torus, the Poincaré bundle \(\mathcal{P}_d\) is a complex line bundle over \(\mathbb{T}^d \times \widehat{\mathbb{Z}}^d\) (where \(\widehat{\mathbb{Z}}^d = \Hom(\mathbb{Z}^d,\mathbb{T})\)). The Fourier-Mukai correspondence is the topological correspondence
\[
\mathbb{T}^d \xleftarrow{\text{pr}_1} (\mathbb{T}^d\times \widehat{\mathbb{Z}}^d, \mathcal{P}_d) \xrightarrow{\text{pr}_2} \widehat{\mathbb{Z}}^d,
\]
which defines an element \([\mathcal{F}_d] \in KK_{-d}(\mathbb{T}^d, \widehat{\mathbb{Z}}^d)\). The Fourier-Mukai transform is the map \([\mathcal{F}_d]\otimes_{\widehat{\mathbb{Z}}^d} : K^{\ast}(\widehat{\mathbb{Z}}^d) \to\) \(K^{\ast -d}(\mathbb{T}^d)\), and the first main result of the paper is a geometric description of this map.
Given a torsion-free discrete group \(G\) with classifying space \(BG\), the Baum-Connes assembly map
\[
\mu : K_{\ast}(BG) \to K_{\ast}(C^{\ast}(G))
\]
is given by
\[
\mu(f) = \mathcal{P}_G\otimes_{C(G)\otimes C^{\ast}(G)}(f\otimes 1_{C^{\ast}(G)})
\]
where \(\mathcal{P}_G \in KK_0(\mathbb{C}, C(BG)\otimes C^{\ast}(G))\) is the class of the Mischenko element (see [\textit{P. Baum} et al., Contemp. Math. 167, 241--291 (1994; Zbl 0830.46061)]). If \(G = \mathbb{Z}^d\), then \(\mathcal{P}_G\) is the class of the Poincaré bundle and the second main result of the paper is a geometric description of the assembly map in this case.
Reviewer: Prahlad Vaidyanathan (Bhopal)Equivariant spectral triples for homogeneous spaces of the compact quantum group \(U_q(2)\)https://zbmath.org/1496.580012022-11-17T18:59:28.764376Z"Guin, Satyajit"https://zbmath.org/authors/?q=ai:guin.satyajit"Saurabh, Bipul"https://zbmath.org/authors/?q=ai:saurabh.bipulSummary: In this article, we study homogeneous spaces \(U_q(2)/_\phi\mathbb{T}\) and \(U_q(2)/_\psi\mathbb{T}\) of the compact quantum group \(U_q(2)\), \(q\in\mathbb{C}\setminus \{0\}\). The homogeneous space \(U_q(2)/_\phi\mathbb{T}\) is shown to be the braided quantum group \(SU_q(2)\). The homogeneous space \(U_q(2)/_\psi\mathbb{T}\) is established as a universal \(C^\ast\)-algebra given by a finite set of generators and relations. Its \(\mathcal{K}\)-groups are computed and two families of finitely summable odd spectral triples, one is \(U_q(2)\)-equivariant and the other is \(\mathbb{T}^2\)-equivariant, are constructed. Using the index pairing, it is shown that the induced Fredholm modules for these families of spectral triples give each element in the \(\mathcal{K}\)-homology group \(K^1(C(U_q(2)/_\psi\mathbb{T}))\).An index formula for groups of isometric linear canonical transformationshttps://zbmath.org/1496.580062022-11-17T18:59:28.764376Z"Savin, Anton"https://zbmath.org/authors/?q=ai:savin.anton-yu"Schrohe, Elmar"https://zbmath.org/authors/?q=ai:schrohe.elmarAuthors' abstract: We define a representation of the unitary group \(U(n)\) by metaplectic operators acting on \( L^2(\mathbb{R}^n) \) and consider the operator algebra generated by the operators of the representation and pseudodifferential operators of Shubin class. Under suitable conditions, we prove the Fredholm property for elements in this algebra and obtain an index formula.
Reviewer: Jan Kurek (Lublin)A 3d gauge theory/quantum \(K\)-theory correspondencehttps://zbmath.org/1496.810852022-11-17T18:59:28.764376Z"Jockers, Hans"https://zbmath.org/authors/?q=ai:jockers.hans"Mayr, Peter"https://zbmath.org/authors/?q=ai:mayr.peterSummary: The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kähler manifold \(X\), which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the \(\mathcal{N} = 2\) supersymmetric 3d gauge theory and the IR limit is given by Givental's permutation equivariant quantum K-theory on \(X\). This gives a one-parameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.