References of "Bruce, Andrew 50022985"
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See detailRepresentations up to Homotopy from Weighted Lie Algebroids
Bruce, Andrew UL; Grabowski, Janusz; Vitagliano, Luca

in Journal of Lie Theory (2018), 28(3), 715-737

Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB-algebroid. There ... [more ▼]

Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB-algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB-algebroids. In this paper we show how this relation generalises to weighted Lie algebroids and in doing so we uncover new and natural examples of higher term representations up to homotopy of Lie algebroids. Moreover, we show how the van Est theorem generalises to weighted objects. [less ▲]

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See detailOn the Concept of a Filtered Bundle
Bruce, Andrew UL; Grabowska, Katarzyna; Grabowski, Janusz

in International Journal of Geometric Methods in Modern Physics (2018), 15

We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of ... [more ▼]

We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalizations of double vector bundles and double affine bundles. Furthermore, we show that the linearization of a filtered bundle — which can be seen as a partial polarization of the admissible changes of local coordinates — is well defined. [less ▲]

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See detailWorkshop on Supergeometry and Applications
Bruce, Andrew UL; Poncin, Norbert UL

Report (2017)

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See detailModular classes of Q-manifolds: a review and some applications
Bruce, Andrew UL

in Archivum Mathematicum (2017)

A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold – which is viewed as the obstruction to the existence of a Q ... [more ▼]

A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold – which is viewed as the obstruction to the existence of a Q-invariant Berezin volume – is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds. [less ▲]

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See detailOn a geometric framework for Lagrangian supermechanics
Bruce, Andrew UL; Grabowska, Katarzyna; Moreno, Giovanni

in Journal of Geometric Mechanics (2017), 9(4), 411-437

We re--examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation ... [more ▼]

We re--examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, we use arbitrary supermanifolds by following the categorical approach to supermanifolds. [less ▲]

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See detailRemarks on Contact and Jacobi Geometry
Bruce, Andrew UL; Grabowska, Katarzyna; Grabowski, Janusz

in Symmetry, Integrability and Geometry: Methods and Applications [=SIGMA] (2017), 13(059), 22

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and ... [more ▼]

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal GL(1,ℝ)-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory. [less ▲]

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See detailRemarks on contact and Jacobi geometry
Bruce, Andrew UL; Grabowska, Katarzyna; Grabowski, Janusz

E-print/Working paper (2016)

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and ... [more ▼]

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and Kirillov algebroids, i.e. homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal GL(1, R)-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. In this sense, the properly understood concept of a Jacobi structure is a specialisation rather than a generalisation of a Poission structure. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, as well as give new insight in the theory. For instance, we describe the structure of Lie groupoids with a compatible principal G-bundle structure and the ‘integrating objects’ for Kirillov algebroids, define canonical contact groupoids, and show that any contact groupoid has a canonical realisation as a contact subgroupoid of the latter. [less ▲]

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