Dirac structures and Nijenhuis operators

Bursztyn, Henrique; Drummond, Thiago; Netto, Clarice

doi:10.48550/arxiv.2109.06330

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A Review Of Multiplicative and Im (1 1) Tensors1

Introduction1

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2022

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“…Example 2.1. Let T ∈ Ω 1 (M, T M) be a (1, 1) tensor on M. Its tangent lift is the linear (1, 1) tensor T tan ∈ Ω 1 (T M, T T M) on T M corresponding to the triple [−, T ] fn , T, T (see, e.g., [9]) where [−, −] fn is the Frölicher-Nijenhuis bracket of vector valued forms.…”

Section: A Review Of Multiplicative and Im (1 1) Tensorsmentioning

confidence: 99%

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Integrating Nijenhuis Structures

Pugliese¹,

Sparano²,

Vitagliano³

2022

Preprint

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A Nijenhuis operator on a manifold M is a (1, 1) tensor N whose Nijenhuistorsion vanishes. A Nijenhuis operator N on M determines a Lie algebroid structure (T M ) N on the tangent bundle T M . In this sense a Nijenhuis operator can be seen as an infinitesimal object. In this paper, we identify its global counterpart. Namely, we show that when the Lie algebroid (T M ) N is integrable, then it integrates to a Lie groupoid equipped with appropriate additional structure responsible for N , and viceversa, the Lie algebroid of a Lie groupoid equipped with such additional structure is of the type (T M ) N for some Nijenhuis operator N . We illustrate our integration result in various examples.

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Section: A Review Of Multiplicative and Im (1 1) Tensorsmentioning

confidence: 99%

“…Finally, we want to mention the recent work of Bursztyn, Drummond, Netto [16,9] on Nijenhuis operators in connections to Lie groupoids, Lie algebroids and related structure (see also [15, ?]). The present paper goes in a complementary direction.…”

Section: Introductionmentioning

confidence: 99%