Dirac Lie groups

Li-Bland, David; Meinrenken, Eckhard

doi:10.4310/ajm.2014.v18.n5.a2

Cited by 19 publications

(23 citation statements)

References 11 publications

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“…11]) is a VB-subgroupoid whenever it has constant rank, see e.g. [4,19]. The analogous result for ∆ 1 follows since Id − K is also a multiplicative projection and ∆ 1 is its kernel.…”

Section: Principal Connectionsmentioning

confidence: 89%

Lie groupoids and the Frölicher-Nijenhuis bracket

Bursztyn

Drummond²

2013

Bull Braz Math Soc, New Series

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“…11]) is a VB-subgroupoid whenever it has constant rank, see e.g. [4,19]. The analogous result for ∆ 1 follows since Id − K is also a multiplicative projection and ∆ 1 is its kernel.…”

Section: Principal Connectionsmentioning

confidence: 89%

Lie groupoids and the Frölicher-Nijenhuis bracket

Bursztyn

Drummond²

2013

Bull Braz Math Soc, New Series

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“…When only h ⊆ d is Lagrangian then the pseudo-connection ∇ is still trivial, and W ⊆ T φ * η G coincides with the Dirac Lie group structure described in [35,36] corresponding to the Dirac Manin triple (d, h; g) ⟨·,·⟩ .…”

Section: Composition Of Courant Relations With Pseudo-dirac Structuresmentioning

confidence: 95%

“…Dirac Lie groups for which multiplication is a morphism of Manin pairs were classified in [36,35] (see also [45,29] for a different setting). There it was shown that the underlying Courant algebroid can be canonically trivialized as an action Courant algebroid A = d × H (see Example 2.8), and the Dirac structure is a constant subbundle E = g × H under this trivialization.…”

Section: Examplesmentioning

confidence: 99%

Pseudo-Dirac structures

Li-Bland

2014

Indagationes Mathematicae

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A Dirac structure is a Lagrangian subbundle of a Courant algebroid, L ⊂ E, which is involutive with respect to the Courant bracket. In particular, L inherits the structure of a Lie algebroid. In this paper, we introduce the more general notion of a pseudo-Dirac structure: an arbitrary subbundle, W ⊂ E, together with a pseudo-connection on its sections, satisfying a natural integrability condition. As a consequence of the definition, W will be a Lie algebroid. Allowing non-isotropic subbundles of E incorporates non-skew tensors and connections into Dirac geometry. Novel examples of pseudo-Dirac structures arise in the context of quasi-Poisson geometry, Lie theory, generalized Kähler geometry, and Dirac Lie groups, among others. Despite their greater generality, we show that pseudo-Dirac structures share many of the key features of Dirac structures. In particular, they behave well under composition with Courant relations.

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“…A VB-groupoid is a groupoid in the category of vector bundles. A CA-groupoid [46,30] is a VB-groupoid A ⇒ A over some Lie groupoid M ⇒ M such that A possesses a Courant algebroid structure (A, , ) which is multiplicative: the graph of the multiplication of A ⇒ A is a Dirac structure inside A × A × A with support on the graph of the multiplication of M. A multiplicative Manin pair (A, E) is a CA-groupoid A ⇒ A and a VB-subgroupoid E ⊂ A such that it is a Dirac structure and E is an LA-groupoid with the restricted Courant bracket. A morphism of multiplicative Manin pairs is a Manin pair morphism given by a Courant relation with support on the graph of a Lie groupoid morphism which is also a VB-subgroupoid inside the product of the CA-groupoids.…”

Section: Multiplicative Manin Pairs and Poisson Groupoidsmentioning

confidence: 99%

Poisson groupoids and moduli spaces of flat bundles over surfaces

Álvarez¹

2021

Preprint

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Let Σ be a compact connected and oriented surface with nonempty boundary and let G be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal G-bundles over Σ which are trivialized at a finite subset of ∂Σ carries a natural quasi-Hamiltonian structure which was introduced by Li-Bland and Severa. By a suitable restriction of the holonomy over ∂Σ and of the gauge action, which is called a decoration of ∂Σ, it is possible to obtain a number of interesting Poisson structures as subquotients of this family of quasi-Hamiltonian structures. In this work we use this quasi-Hamiltonian structure to construct Poisson and symplectic groupoids in a systematic fashion by means of two observations:(1) gluing two copies of the same decorated surface along suitable subspaces of their boundaries determines a groupoid structure on the moduli space associated to the new surface, this procedure can be iterated by gluing four copies of the same surface, thereby inducing a double Poisson groupoid structure;(2) on the other hand, we can suppose that G is a Lie 2-group, then the groupoid structure on G descends to a groupoid structure on the moduli space of flat G-bundles over Σ. These two observations can be combined to produce up to three distinct and compatible groupoid structures on the associated moduli spaces. We illustrate these methods by considering symplectic groupoids over Bruhat cells, twisted moduli spaces and Poisson 2-groups besides the classical examples. Contents

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Dirac Lie groups

Cited by 19 publications

References 11 publications

Lie groupoids and the Frölicher-Nijenhuis bracket

Lie groupoids and the Frölicher-Nijenhuis bracket

Pseudo-Dirac structures

Poisson groupoids and moduli spaces of flat bundles over surfaces

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