Covariance and geometrical invariance in *quantization

Arnal, Didier; Cortet, J. C.; Molin, Paul; Pinczon, Georges

doi:10.1063/1.525703

Cited by 52 publications

(53 citation statements)

References 5 publications

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“…x, g (1) (AS(g (2) ) y) = g (2) S(g (3) ) (2) S(g (3) ) * (1) (A * g * (1) x), y = S(g (2) ) * (A * g * (1) x), y , using twice (2.13) and the fact that A is adjointable as well as S ⊗ S • op = • S. This shows that g A is indeed adjointable with adjoint given by (2) ) * A * g (1) …”

Section: Hopf * -Algebras and * -Actionsmentioning

confidence: 99%

The H-covariant strong Picard groupoid

Jansen

Waldmann

2006

Journal of Pure and Applied Algebra

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The notion of H-covariant strong Morita equivalence is introduced for * -algebras over C = R(i) with an ordered ring R which are equipped with a * -action of a Hopf * -algebra H. This defines a corresponding H-covariant strong Picard groupoid which encodes the entire Morita theory. Dropping the positivity conditions one obtains H-covariant * -Morita equivalence with its H-covariant * -Picard groupoid. We discuss various groupoid morphisms between the corresponding notions of the Picard groupoids. Moreover, we realize several Morita invariants in this context as arising from actions of the H-covariant strong Picard groupoid. Crossed products and their Morita theory are investigated using a groupoid morphism from the H-covariant strong Picard groupoid into the strong Picard groupoid of the crossed products.

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Section: Hopf * -Algebras and * -Actionsmentioning

confidence: 99%

The H-covariant strong Picard groupoid

Jansen

Waldmann

2006

Journal of Pure and Applied Algebra

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“…the early work [1]. The resulting general notion of a quantum momentum map is due to [26] but was already used in examples in e.g.…”

Section: Introductionmentioning

confidence: 97%

Classification of Equivariant Star Products on Symplectic Manifolds

Reichert

Waldmann

2016

Lett Math Phys

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In this note we classify invariant star products with quantum momentum maps on symplectic manifolds by means of an equivariant characteristic class taking values in the equivariant cohomology. We establish a bijection between the equivalence classes and the formal series in the second equivariant cohomology, thereby giving a refined classification which takes into account the quantum momentum map as well.Comment: 15 pages, no figure

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“…We shall always reserve the symbols u k and v l to denote functions as in (1). Moreover, Karabegov considers locally defined formal series of one-forms α,…”

Section: Karabegov's Description and Characterization Of Star Productmentioning

confidence: 99%

“…that satisfies r(g)J(ξ) = J(Ad(g)ξ) for all g ∈ G and for all ξ ∈ g, clearly defines a quantum momentum mapping which is called a G-equivariant quantum momentum mapping. Also recall the definition of a strongly invariant star product from [1]: Let J 0 be a classical momentum mapping for the action r resp. ̺.…”

Section: Quantum Momentum Mappings For Invariant Star Products Of Wicmentioning

confidence: 99%

“…In the framework of deformation quantization, however, this incorporation can at least be formulated very naturally as it has already been indicated in the pioneering articles [2] by Bayen, Flato, Frønsdal, Lichnerowicz, and Sternheimer. Various notions of invariance of star products with respect to actions of Lie groups and Lie algebras were introduced and discussed by Arnal, Cortet, Molin, and Pinczon in [1]. Previously, the existence of G-invariant symplectic connections has been related to that of certain G-invariant star products by Lichnerowicz in [20].…”

Section: Introductionmentioning

confidence: 99%

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Invariant Star Products of Wick Type: Classification and Quantum Momentum Mappings

Müller-Bahns

Neumaier

2004

Letters in Mathematical Physics

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We extend our investigations on g-invariant Fedosov star products and quantum momentum mappings [22] to star products of Wick type on pseudo-Kähler manifolds. Star products of Wick type can be completely characterized by a local description as given by Karabegov in [14] for star products with separation of variables. We separately treat the action of a Lie group] by (pull-backs with) diffeomorphisms and the action of a Lie algebra g on C ∞ (M ) [[ν]] by (Lie derivatives with respect to) vector fields. Within Karabegov's framework, we prove necessary and sufficient conditions for a given star product of Wick type to be invariant in the respective sense. Moreover, our results yield a complete classification of invariant star products of Wick type. We also prove a necessary and sufficient condition for (the Lie derivative with respect to) a vector field to be even a quasi-inner derivation of a given star product of Wick type. We then transfer our former results about quantum momentum mappings for g-invariant Fedosov star products to the case of invariant star products of Wick type.

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Covariance and geometrical invariance in *quantization

Cited by 52 publications

References 5 publications

The H-covariant strong Picard groupoid

The H-covariant strong Picard groupoid

Classification of Equivariant Star Products on Symplectic Manifolds

Invariant Star Products of Wick Type: Classification and Quantum Momentum Mappings

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