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Almost Commutative Q-algebras and Derived brackets Bruce, Andrew in Journal of Noncommutative Geometry (in press) We introduce the notion of almost commutative Q-algebras and demonstrate how the derived bracket formalism of Kosmann-Schwarzbach generalises to this setting. In particular, we construct ‘almost ... [more ▼] We introduce the notion of almost commutative Q-algebras and demonstrate how the derived bracket formalism of Kosmann-Schwarzbach generalises to this setting. In particular, we construct ‘almost commutative Lie algebroids’ following Vaintrob’s Q-manifold understanding of classical Lie algebroids. We show that the basic tenets of the theory of Lie algebroids carry over verbatim to the almost commutative world. [less ▲] Detailed reference viewed: 21 (0 UL)Connections adapted to non-negatively graded structure Bruce, Andrew in International Journal of Geometric Methods in Modern Physics (2018) Graded bundles are a particularly nice class of graded manifolds and represent a natural generalization of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids, we ... [more ▼] Graded bundles are a particularly nice class of graded manifolds and represent a natural generalization of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids, we define the notion of a weighted A-connection on a graded bundle. In a natural sense weighted A-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear A-connections are adapted to the structure of a vector bundle. This notion generalizes directly to multi-graded bundles and in particular we present the notion of a bi-weighted A-connection on a double vector bundle. We prove the existence of such adapted connections and use them to define (quasi-)actions of Lie algebroids on graded bundles. [less ▲] Detailed reference viewed: 34 (3 UL)Products in the category of Z_2^n manifolds Bruce, Andrew ; Poncin, Norbert E-print/Working paper (2018) Detailed reference viewed: 51 (6 UL)Functional analytic issues in Z_2 ^n Geometry Bruce, Andrew ; Poncin, Norbert E-print/Working paper (2018) Detailed reference viewed: 39 (3 UL)Representations up to Homotopy from Weighted Lie Algebroids Bruce, Andrew ; ; 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 ▲] Detailed reference viewed: 52 (1 UL)On the Concept of a Filtered Bundle Bruce, Andrew ; ; 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 ▲] Detailed reference viewed: 38 (12 UL)Workshop on Supergeometry and Applications Bruce, Andrew ; Poncin, Norbert Report (2017) Detailed reference viewed: 81 (9 UL)Modular classes of Q-manifolds: a review and some applications Bruce, Andrew 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 ▲] Detailed reference viewed: 13 (3 UL)On a geometric framework for Lagrangian supermechanics Bruce, Andrew ; ; 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 ▲] Detailed reference viewed: 17 (3 UL)Remarks on Contact and Jacobi Geometry Bruce, Andrew ; ; 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 ▲] Detailed reference viewed: 21 (4 UL)Remarks on contact and Jacobi geometry Bruce, Andrew ; ; 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 ▲] Detailed reference viewed: 55 (5 UL) |
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