Integration of quadratic Lie algebroids to Riemannian Cartan–Lie groupoids

Kotov, Alexei; Strobl, Thomas

doi:10.1007/s11005-018-1048-1

Cited by 4 publications

(9 citation statements)

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“…On the other hand, we will not require the compatibility of this Lie algebroid structure with the connection except if otherwise stated. One option for a passage from a Killing anchored bundle to a Killing Cartan Lie algebroid is explained in [16] Lemma 2. By means of g, we can view the anchor ρ ∈ Γ(A * ⊗ T M) as a section of A * ⊗ T * M, which we denote by ρ.…”

Section: Example 5 ([3]) Any Torsion-free Connection On T M Gives Ris...mentioning

confidence: 99%

“…Second, if one has a Cartan-Lie algebroid (A, ∇) together with a metric g on its base, compatible in the sense of equation ( 1), and a fiber metric A g, compatible with (A, ∇) in the sense of A ∇ * A g = 0 (with the A-connection introduced above), one may see that these data give the appropriate generalisation of a quadratic Lie algebra to the setting of Lie algebroids. In [16] we determine the necessary and sufficient conditions for this to integrate to a Riemannian groupoid for the case that the Lie algebroid A itself is integrable to a groupoid. Interestingly, there are obstructions to this integration in general.…”

Section: Introductionmentioning

confidence: 99%

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Lie algebroids, gauge theories, and compatible geometrical structures

Kotov

Strobl

2019

Rev. Math. Phys.

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The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper is supposed to analyze these compatibilities from a mathematical perspective.In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove furthermore that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized Riemannian structure.

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Section: Example 5 ([3]) Any Torsion-free Connection On T M Gives Ris...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lie algebroids, gauge theories, and compatible geometrical structures

Kotov

Strobl

2019

Rev. Math. Phys.

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“…Even when G is a Lie groupoid, the Γ -inertia groupoid Λ Γ G is not smooth and has nontrivial local topology; see [15,16] for a thorough description of the case Γ " Z. Using the notion of an ℓ-metric introduced by del Hoyo and Fernandes [11,12] as well as its extension to the case of Cartan-Lie groupoids [3,25], we investigate the induced Riemannian structures on smooth subsets of the Γ -inertia space. Using the fact that the isotropy groups G x…”

Section: Introductionmentioning

confidence: 99%

Orbifold Euler characteristics of non‐orbifold groupoids

Farsi

Seaton

2022

Journal of London Math Soc

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For a finitely presented discrete group Γ, we introduce two generalizations of the orbifold Euler characteristic and Γ-orbifold Euler characteristic to a class of proper topological groupoids large enough to include all cocompact proper Lie groupoids. The Γ-Euler characteristic is defined as an integral with respect to the Euler characteristic over the orbit space of the groupoid, and the Γ-inertia Euler characteristic is the usual Euler characteristic of the Γ-inertia space associated to the groupoid. A key ingredient is the application of o-minimal structures to study orbit spaces of topological groupoids. Our main result is that the Γ-Euler characteristic and Γinertia Euler characteristic coincide and generalize the higher order orbifold Euler characteristics of Gusein-Zade, Luengo, and Melle-Hernández from the case of a translation groupoid by a compact Lie group and Γ = ℤ 𝓁 . By realizing the Γ-Euler characteristic as the usual Euler characteristic of a topological space, we demonstrate that it is Morita invariant in the category of topological groupoids and satisfies familiar properties of the classical Euler characteristic. We give an additional formulation of the Γ-Euler characteristic for a cocompact proper Lie groupoid in terms of a finite covering by orbispace charts. In the case that the groupoid is an abelian extension of a translation groupoid by a bundle

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“…The Hamiltonian equation (1) explains well the heuristic meaning: When going along the leaves of the foliation, only those components of the (inverse) metric which go along the leaves may change. We refer to [28,29] for a purely geometrical analysis of this and similar compatibility equations; there it is shown, e.g., that for an arbitrary Lie algebroid the condition (18) implies that the (possibly singular) foliation on M induced by (E, ρ) is Riemannian with respect to g.…”

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confidence: 99%

“…For Dirac structures projectable to T M, the BFV-functional takes the minimal form (40)-with Φ a replaced by J a , resulting from choosing a local basis s a in the (small) Dirac structure. Otherwise there are finitely many dependences and one needs further global ghosts to satisfy the test (29) in smooth cases, cf., e.g., [34]. A Lagrangian leading to this system is the Dirac sigma model [25].…”

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confidence: 99%

On the Relation of Lie Algebroids to Constrained Systems and their BV/BFV Formulation

Ikeda

Strobl

2019

Ann. Henri Poincaré

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We observe that a system of irreducible, fiber-linear, first class constraints on T * M is equivalent to the definition of a foliation Lie algebroid over M . The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1], Q) of the Lie algebroid to its cotangent bundle T * E[1]. Affine deformations of the constraints are parametrized by the first Lie algebroid cohomology H 1 Q and lead to irreducible constraints also for much more general Lie algebroids such as Dirac structures; the modified BFV function follows by the addition of a representative of the deformation charge.Adding a Hamiltonian to the system corresponds to a metric g on M . Evolution invariance of the constraint surface introduces a connection ∇ on E and one reobtains the compatibility of g with (E, ρ, ∇) found previously in the literature. The covariantization of the Hamiltonian to a function on T * E[1] serves as a BFV-Hamiltonian, iff, in addition, this connection is compatible with the Lie algebroid structure, turning (E, ρ, [·, ·], ∇) into a Cartan-Lie algebroid. The BV formulation of the system is obtained from BFV by a (time-dependent) AKSZ procedure.

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Integration of quadratic Lie algebroids to Riemannian Cartan–Lie groupoids

Cited by 4 publications

References 28 publications

Lie algebroids, gauge theories, and compatible geometrical structures

Lie algebroids, gauge theories, and compatible geometrical structures

Orbifold Euler characteristics of non‐orbifold groupoids

On the Relation of Lie Algebroids to Constrained Systems and their BV/BFV Formulation

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