Recent zbMATH articles in MSC 53D17https://zbmath.org/atom/cc/53D172021-06-15T18:09:00+00:00WerkzeugInfinitesimal automorphisms of VB-groupoids and algebroids.https://zbmath.org/1460.530712021-06-15T18:09:00+00:00"Esposito, Chiara"https://zbmath.org/authors/?q=ai:esposito.chiara"Vitagliano, Luca"https://zbmath.org/authors/?q=ai:vitagliano.luca"Tortorella, Alfonso Giuseppe"https://zbmath.org/authors/?q=ai:tortorella.alfonso-giuseppeSummary: VB-groupoids and algebroids are \textit{vector bundle objects} in the categories of Lie groupoids and Lie algebroids, respectively, and they are related via the \textit{Lie functor}. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation VB-groupoid/algebroid.Poisson-Lie algebras and singular symplectic forms associated to corank 1 type singularities.https://zbmath.org/1460.530722021-06-15T18:09:00+00:00"Fukuda, T."https://zbmath.org/authors/?q=ai:fukuda.takuo"Janeczko, S."https://zbmath.org/authors/?q=ai:janeczko.stanislawSummary: We show that there exists a natural Poisson-Lie algebra associated to a singular symplectic structure \(\omega \). We construct Poisson-Lie algebras for the Martinet and Roussarie types of singularities. In the special case when the singular symplectic structure is given by the pullback from the Darboux form, \( \omega=F^*\omega_0\), this Poisson-Lie algebra is a basic symplectic invariant of the singularity of the smooth mapping \(F\) into the symplectic space \((\mathbb{R}^{2n},\omega_0)\). The case of \(A_k\) singularities of pullbacks is considered, and Poisson-Lie algebras for \(\Sigma_{2,0}\), \(\Sigma_{2,2,0}^{\mathrm{e}}\) and \(\Sigma_{2,2,0}^{\mathrm{h}}\) stable singularities of 2-forms are calculated.Bott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces.https://zbmath.org/1460.530732021-06-15T18:09:00+00:00"Lu, Jiang-Hua"https://zbmath.org/authors/?q=ai:lu.jiang-hua|lu.jianghua"Yu, Shizhuo"https://zbmath.org/authors/?q=ai:yu.shizhuoSummary: Let \(G\) be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous \(G\)-spaces \(G/Q\), we construct a finite atlas \(\mathcal{A}_{BS} (G/Q)\) on \(G/Q\), called the \textit{Bott-Samelson atlas}, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on \(G/Q\). We also show that the standard Poisson structure \(\pi_G/Q\) on \(G/Q\) is presented, in each of the coordinate charts of \(\mathcal{A}_{BS} (G/Q)\), as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making \((G/Q, \pi_G/Q, \mathcal{A}_{BS} (G/Q))\) into a \textit{Poisson-Ore variety}. In addition, all coordinate functions in the Bott-Samelson atlas are shown to have complete Hamiltonian flows with respect to the Poisson structure \(\pi_G/Q\). Examples of \(G/Q\) include \(G\) itself, \(G/T\), \(G/B\), and \(G/N\), where \(T \subset G\) is a maximal torus, \(B \subset G\) a Borel subgroup, and \(N\) the uniradical of \(B\).Odd transgression for Courant algebroids.https://zbmath.org/1460.530702021-06-15T18:09:00+00:00"Bressler, Paul"https://zbmath.org/authors/?q=ai:bressler.paul"Rengifo, Camilo"https://zbmath.org/authors/?q=ai:rengifo.camiloSummary: The ``odd transgression'' introduced in [the authors, Lett. Math. Phys. 108, No. 9, 2099--2137 (2018; Zbl 1397.53092)] is applied to construct and study the inverse image functor in the theory of Courant algebroids.BV and BFV for the H-twisted Poisson sigma model.https://zbmath.org/1460.810512021-06-15T18:09:00+00:00"Ikeda, Noriaki"https://zbmath.org/authors/?q=ai:ikeda.noriaki"Strobl, Thomas"https://zbmath.org/authors/?q=ai:strobl.thomasSummary: We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form \(H\). There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using an auxiliary connection \(\nabla\) on the target manifold \(M\), we obtain alternative, purely geometrical expressions without the use of superfields, which are new also for the ordinary PSM and promise adaptations to other Lie algebroid-based gauge theories: The BV functional, in particular, is the sum of the classical action, the Hamiltonian lift of the (only on-shell nilpotent) BRST differential, and a term quadratic in the antifields which is essentially the basic curvature and measures the compatibility of \(\nabla\) with the Lie algebroid structure on \(T^*M\). We finally construct a \(\operatorname{Diff}(M)\)-equivariant isomorphism between the two BV formulations.A linear algebraic setting for Jacobi structures.https://zbmath.org/1460.530652021-06-15T18:09:00+00:00"Cioroianu, Eugen-Mihăiţă"https://zbmath.org/authors/?q=ai:cioroianu.eugen-mihaita"Vizman, Cornelia"https://zbmath.org/authors/?q=ai:vizman.corneliaThe properties that are of linear algebraic nature of Poisson, Jacobi and contact geometries are considered.
The theory of the linear version of so-called contact dual pairs is developed. In particular, it is proved that in a full linear contact dual pair, the characteristic subspace of the two Jacobi vector spaces are either both odd-dimensional contact, or both even-dimensional locally conformally symplectic.
Reviewer: Mihail Banaru (Smolensk)