Recent zbMATH articles in MSC 53D17https://www.zbmath.org/atom/cc/53D172021-04-16T16:22:00+00:00WerkzeugShifted derived Poisson manifolds associated with Lie pairs.https://www.zbmath.org/1456.530652021-04-16T16:22:00+00:00"Bandiera, Ruggero"https://www.zbmath.org/authors/?q=ai:bandiera.ruggero"Chen, Zhuo"https://www.zbmath.org/authors/?q=ai:chen.zhuo"Stiénon, Mathieu"https://www.zbmath.org/authors/?q=ai:stienon.mathieu"Xu, Ping"https://www.zbmath.org/authors/?q=ai:xu.pingSummary: We study the shifted analogue of the ``Lie-Poisson'' construction for \(L_\infty\) algebroids and we prove that any \(L_\infty\) algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras. As an application, we prove that, given a Lie pair \((L, A)\), the space \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A))\) admits a degree \((+1)\) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley-Eilenberg differential \(d_A^{\mathrm{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\rightarrow\Omega^{\bullet+1}_A(\Lambda^\bullet (L/A))\) as unary \(L_\infty\) bracket. This degree \((+1)\) derived Poisson algebra structure on \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\) is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley-Eilenberg hypercohomology \(\mathbb{H}(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A)),d_A^{\mathrm{Bott}})\) admits a canonical Gerstenhaber algebra structure.The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps.https://www.zbmath.org/1456.530662021-04-16T16:22:00+00:00"Cruz, Inês"https://www.zbmath.org/authors/?q=ai:cruz.ines"Mena-Matos, Helena"https://www.zbmath.org/authors/?q=ai:mena-matos.helena"Sousa-Dias, Esmeralda"https://www.zbmath.org/authors/?q=ai:sousa-dias.esmeraldaSummary: We consider a family of birational maps \(\varphi_k\) in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family \(\varphi_k\) using Poisson geometry tools, namely the properties of the restrictions of the maps \(\varphi_k\) and their fourth iterate \(\varphi^{(4)}_k\) to the symplectic leaves of an appropriate Poisson manifold \((\mathbb{R}^4_+, P)\). These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product \(SL(2, \mathbb{Z})\ltimes\mathbb{R}^2\). The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for \(\varphi_k\) characterized by the parameter values \(k = 1\), \(k = 2\) and \(k\geq 3\).The method of averaging for Poisson connections on foliations and its applications.https://www.zbmath.org/1456.530642021-04-16T16:22:00+00:00"Avendaño-Camacho, Misael"https://www.zbmath.org/authors/?q=ai:avendano-camacho.misael"Hasse-Armengol, Isaac"https://www.zbmath.org/authors/?q=ai:hasse-armengol.isaac"Velasco-Barreras, Eduardo"https://www.zbmath.org/authors/?q=ai:velasco-barreras.eduardo"Vorobiev, Yury"https://www.zbmath.org/authors/?q=ai:vorobiev.yuriiSummary: On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.On LA-Courant algebroids and Poisson Lie 2-algebroids.https://www.zbmath.org/1456.580042021-04-16T16:22:00+00:00"Jotz Lean, M."https://www.zbmath.org/authors/?q=ai:jotz.madeleineT.J. Courant discovered a skew-symmetric Lie bracket on \(TM \oplus T^* M\). The more general structure of a Courant algebroid, links the study of constrained Hamiltonian systems with generalised complex geometry. They were studied extensively throughout the 1990s by Zhang-Ju Liu, Alan Weinstein and Ping Xu, as well as Severa and Roytenberg. The associated integrability problem is an open question to this day.
To this end, it is important to understand better these structures. Courant algebroids are ``higher'' geometric structures. This can be made precise in the following ways: Roytenberg and Severa (independently) understood them in a graded sense, namely they described them as symplectic Lie 2-algebroids. On the other hand, Courant's example fits into \textit{K. C. H. Mackenzie}'s study of multiple structures, in particular it is an example of a double Lie algebroid [J. Reine Angew. Math. 658, 193--245 (2011; Zbl 1246.53112)]. \textit{D. Li-Bland} in his PhD thesis [LA-Courant algebroids and their applications. \url{arXiv:1204.2796}] introduced a more general class of Courant algebroids (LA-Courant algebroids) which are Courant algebroid structures in the category of vector bundles. They too fit in the multiple structures studied by Mackenzie.
The paper under review studies the correspondence between LA-Courant algebroids and Poisson Lie 2-algebroids (the latter generalize symplectic Lie 2-algebroids), using the author's earlier results on split Lie 2-algebroids and self-dual 2-representations.
Reviewer: Iakovos Androulidakis (Athína)Reduction of Nambu-Poisson manifolds by regular distributions.https://www.zbmath.org/1456.530672021-04-16T16:22:00+00:00"Das, Apurba"https://www.zbmath.org/authors/?q=ai:das.apurbaSummary: The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by \textit{R. Ibáñez} et al. [Rep. Math. Phys. 42, No. 1-2, 71--90 (1998; Zbl 0931.37024)]. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduction theorem for Nambu-Poisson manifolds. Finally, we define gauge transformations of Nambu-Poisson structures and show that these transformations commute with the reduction procedure.Affine structures on Lie groupoids.https://www.zbmath.org/1456.220012021-04-16T16:22:00+00:00"Lang, Honglei"https://www.zbmath.org/authors/?q=ai:lang.honglei"Liu, Zhangju"https://www.zbmath.org/authors/?q=ai:liu.zhangju"Sheng, Yunhe"https://www.zbmath.org/authors/?q=ai:sheng.yunheAuthors' abstract: We study affine structures on a Lie groupoid, including affine \(k\)-vector fields, \(k\)-forms and \((p, q)\)-tensors. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields with the Schouten bracket and the space of affine vector-valued forms with the Frölicher-Nijenhuis bracket are graded strict Lie 2-algebras, and affine \((1, 1)\)-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.
Reviewer: Iakovos Androulidakis (Athína)