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Journal of Geometric Mechanics

June 2022 , Volume 14 , Issue 2

Special issue in memory of Kirill C. H. Mackenzie: Part II

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Preface to special issue in honor of Kirill C. H. Mackenzie: Part Ⅱ
Iakovos Androulidakis, Henrique Burzstyn, Juan-Carlos Marrero and Alan Weinstein
2022, 14(2): i-ii doi: 10.3934/jgm.2022016 +[Abstract](150) +[HTML](64) +[PDF](110.43KB)
Poisson double structures
Henrique Bursztyn, Alejandro Cabrera and Matias del Hoyo
2022, 14(2): 151-178 doi: 10.3934/jgm.2021029 +[Abstract](552) +[HTML](188) +[PDF](567.09KB)

We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these objects are related by differentiation and integration. We use these results to revisit Lie 2-bialgebras by means of Poisson double structures.

Constrained systems, generalized Hamilton-Jacobi actions, and quantization
Alberto S. Cattaneo, Pavel Mnev and Konstantin Wernli
2022, 14(2): 179-272 doi: 10.3934/jgm.2022010 +[Abstract](206) +[HTML](107) +[PDF](1215.17KB)

Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton–Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern–Simons theory, where the HJ action turns out to be the gauged Wess–Zumino–Witten action). Perturbative quantization, limited in this note to finite-dimensional targets, is performed in the framework of the Batalin–Vilkovisky (BV) formalism in the bulk and of the Batalin–Fradkin–Vilkovisky (BFV) formalism at the endpoints. As a sanity check of the method, it is proved that the semiclassical contribution of the physical part of the evolution operator is still given by the HJ action. Several examples are computed explicitly. In particular, it is shown that the toy model for nonabelian Chern–Simons theory and the toy model for 7D Chern–Simons theory with nonlinear Hitchin polarization do not have quantum corrections in the physical part (the extension of these results to the actual cases is discussed in the companion paper [21]). Background material for both the classical part (symplectic geometry, generalized generating functions, HJ actions, and the extension of these concepts to infinite-dimensional manifolds) and the quantum part (BV-BFV formalism) is provided.

Modular class of Lie $ \infty $-algebroids and adjoint representations
Raquel Caseiro and Camille Laurent-Gengoux
2022, 14(2): 273-305 doi: 10.3934/jgm.2022008 +[Abstract](189) +[HTML](72) +[PDF](604.47KB)

We study the modular class of \begin{document}$ Q $\end{document}-manifolds, and in particular of negatively graded Lie \begin{document}$ \infty $\end{document}-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie \begin{document}$ \infty $\end{document}-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie \begin{document}$ \infty $\end{document}-algebroids and their \begin{document}$ Q $\end{document}-manifold equivalent, which we hope to be of use for future reference.

Quotients of double vector bundles and multigraded bundles
Eckhard Meinrenken
2022, 14(2): 307-329 doi: 10.3934/jgm.2021027 +[Abstract](602) +[HTML](246) +[PDF](478.79KB)

We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for weighted submanifolds, as well as for pairs of submanifolds with clean intersection.

The evolution equation: An application of groupoids to material evolution
Víctor Manuel Jiménez and Manuel de León
2022, 14(2): 331-348 doi: 10.3934/jgm.2022001 +[Abstract](460) +[HTML](153) +[PDF](449.26KB)

The aim of this paper is to study the evolution of a material point of a body by itself, and not the body as a whole. To do this, we construct a groupoid encoding all the intrinsic properties of the material point and its characteristic foliations, which permits us to define the evolution equation. We also discuss phenomena like remodeling and aging.

Towards a 2-dimensional notion of holonomy
Ronald Brown and İlhan İçen
2022, 14(2): 349-375 doi: 10.3934/jgm.2022011 +[Abstract](160) +[HTML](105) +[PDF](608.41KB)

Previous work (Pradines, C.R. Acad. Sci. Paris 263 (1966) 907, Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the \begin{document}$ 2 $\end{document}-dimensional information.

Kirill Mackenzie, Bangor and holonomy
Ronald Brown
2022, 14(2): 377-380 doi: 10.3934/jgm.2022012 +[Abstract](171) +[HTML](56) +[PDF](286.26KB)

2021 Impact Factor: 0.737
5 Year Impact Factor: 0.713
2021 CiteScore: 1.3



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