Connections adapted to non-negatively graded structures

Bruce, Andrew James

doi:10.1142/s021988781950021x

Cited by 3 publications

(3 citation statements)

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“…(4) Linear connections can be reformulated as odd vector fields on a particular bi-graded supermanifold build from the initial anchored vector bundle, see [4] for details. We will avoid graded/weighted geometry in this note and stick to a more classical presentation.…”

Section: Heaps Of Connectionsmentioning

confidence: 99%

The Ternary Structure of Lie Algebroid Connections

Bruce¹

2023

Preprint

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Section: Heaps Of Connectionsmentioning

confidence: 99%

The Ternary Structure of Lie Algebroid Connections

Bruce¹

2023

Preprint

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“…For an overview of connections in classical and quantum field theory the reader may consult [30]. Over the years there have been many generalisations of a connection on a manifold given in the literature, including the generalisation to Lie algebroids, Courant algebroids (see [17]) and connections adapted to nonnegatively graded manifolds (see [7]), to name a few. In the noncommutative setting, we have, for example, linear connections on bimodules over almost commutative algebras (see [11]).…”

Section: Introductionmentioning

confidence: 99%

Odd connections on supermanifolds: existence and relation with affine connections

Bruce¹,

Grabowski²

2020

J. Phys. A: Math. Theor.

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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 N = 1 super-Minkowski spacetime admits a natural odd connection.

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“…Linear connections can be reformulated as odd vector fields on a particular bi-graded supermanifold built from the initial anchored vector bundle (see [21] for details). We avoid graded/weighted geometry in this note and stick to a more classical presentation.…”

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confidence: 99%

Heaps of Linear Connections and Their Endomorphism Truss

Bruce

2024

Symmetry

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We examine the heap of linear connections on anchored vector bundles and Lie algebroids. Naturally, this covers the example of affine connections on a manifold. We present some new interpretations of classical results via this ternary structure of connections. Endomorphisms of linear connections are studied, and their ternary structure, in particular the endomorphism truss, is explicitly presented. We remark that the use of ternary structures in differential geometry is novel and that the endomorphism truss of linear connections provides a concrete geometric example of a truss.

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Connections adapted to non-negatively graded structures

Cited by 3 publications

References 44 publications

The Ternary Structure of Lie Algebroid Connections

The Ternary Structure of Lie Algebroid Connections

Odd connections on supermanifolds: existence and relation with affine connections

Heaps of Linear Connections and Their Endomorphism Truss

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