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Odd connections on supermanifolds: existence and relation with affine connections Bruce, Andrew ; in Journal of Physics. A, Mathematical and Theoretical (2020), 53(45), 455203 The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in ... [more ▼] The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A special class of such generalised connections, referred to as odd connections in this paper, have torsion and curvature tensors. Part of the structure is an odd involution of the tangent bundle of the supermanifold and this puts drastic restrictions on the supermanifolds that admit odd connections. In particular, they must have equal number of even and odd dimensions. Amongst other results, we show that an odd connection is defined, up to an odd tensor field of type (1, 2), by an affine connection and an odd endomorphism of the tangent bundle. Thus, the theory of odd connections and affine connections are not completely separate theories. As an example relevant to physics, it is shown that $\mathcal{N}=1$ super-Minkowski spacetime admits a natural odd connection. [less ▲] Detailed reference viewed: 27 (2 UL)Riemannian Structures on Z 2 n -Manifolds Bruce, Andrew ; in Mathematics (2020), 8(9), 1469 Very loosely, Zn2-manifolds are ‘manifolds’ with Zn2-graded coordinates and their sign rule is determined by the scalar product of their Zn2-degrees. A little more carefully, such objects can be ... [more ▼] Very loosely, Zn2-manifolds are ‘manifolds’ with Zn2-graded coordinates and their sign rule is determined by the scalar product of their Zn2-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Zn2-manifold, i.e., a Zn2-manifold equipped with a Riemannian metric that may carry non-zero Zn2-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Zn2-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry [less ▲] Detailed reference viewed: 23 (0 UL)Pre-Courant algebroids Bruce, Andrew ; in Journal of Geometry and Physics (2019), 142 Pre-Courant algebroids are ‘Courant algebroids’ without the Jacobi identity for the Courant–Dorfman bracket. We examine the corresponding supermanifold description of pre-Courant algebroids and some ... [more ▼] Pre-Courant algebroids are ‘Courant algebroids’ without the Jacobi identity for the Courant–Dorfman bracket. We examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof. In particular, we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. We give the definition of (sub-)Dirac structures and study the naïve quasi-cochain complex within the setting of supergeometry. Moreover, the framework of supermanifolds allows us to economically define and work with pre-Courant algebroids equipped with a compatible non-negative grading. VB-Courant algebroids are natural examples of what we call weighted pre-Courant algebroids and our approach drastically simplifies working with them. [less ▲] Detailed reference viewed: 38 (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: 57 (12 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: 60 (1 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: 86 (4 UL)Splitting theorem for Z_2^n-supermanifolds ; ; Poncin, Norbert in Journal of Geometry & Physics (2016), 110 Detailed reference viewed: 126 (20 UL)The category of Z_2^n-supermanifolds ; ; Poncin, Norbert in Journal of Mathematical Physics (2016), 57(7), 16 Detailed reference viewed: 255 (42 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: 122 (5 UL)Integration on colored supermanifolds ; Kwok, Stephen ; Poncin, Norbert E-print/Working paper (2016) Detailed reference viewed: 75 (6 UL)Z_2^n-Supergeometry II: Batchelor-Gawedzki Theorem Covolo, Tiffany ; ; Poncin, Norbert E-print/Working paper (2014) Detailed reference viewed: 107 (15 UL)Z_2^n-Supergeometry I: Manifolds and Morphisms Covolo, Tiffany ; ; Poncin, Norbert E-print/Working paper (2014) Detailed reference viewed: 284 (73 UL)Lie superalgebras of differential operators ; ; Poncin, Norbert in Journal of Lie Theory (2013), 23(1), 35--54 Detailed reference viewed: 94 (9 UL)The supergeometry of Loday algebroids ; Khudaverdyan, David ; Poncin, Norbert in Journal of Geometric Mechanics (2013), 5(2), 185--213 Detailed reference viewed: 209 (26 UL)Geometric structures encoded in the Lie structure of an Atiyah algebroid ; ; Poncin, Norbert in Transformation Groups (2011), 16(1), 137--160 Detailed reference viewed: 149 (2 UL)The Lie superalgebra of a supermanifold ; ; Poncin, Norbert in Journal of Lie Theory (2010), 20(4), 739--749 Detailed reference viewed: 97 (5 UL)On quantum and classical Poisson algebras ; Poncin, Norbert in Banach Center Publications (2007), 76 Detailed reference viewed: 97 (2 UL)Derivations of the Lie algebras of differential operators ; Poncin, Norbert in Indagationes Mathematicae (2005), 16(2), 181--200 Detailed reference viewed: 108 (1 UL)Lie algebraic characterization of manifolds ; Poncin, Norbert in Central European Journal of Mathematics (2004), 2(5), 811--825 Detailed reference viewed: 111 (6 UL)Automorphisms of quantum and classical Poisson algebras ; Poncin, Norbert in Compositio Mathematica (2004), 140(2), 511-527 Detailed reference viewed: 113 (8 UL) |
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