Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces

Bieliavsky, Pierre; Claessens, Laurent; Sternheimer, Daniel; Voglaire, Yannick

doi:10.1090/conm/450/08731

Cited by 7 publications

(10 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where (V, ω 0 ) is the symplectic vector space attached to S. In particular, the action of B ′ leaves invariant the two-point kernel K θ,τ on S × S. Iterating the above observation at the level of B ′ and translating the "extension Lemma" in [6] within the present framework, we obtain:…”

Section: Star-products On Normal J-groupsmentioning

confidence: 80%

“…where a, t 1 , t 2 , ℓ ∈ R and v 1 , v 2 ∈ V . The group law ofG in these coordinates then reads: 6) and the inversion map is given by:…”

Section: The Transvection Quadruple Of An Elementary Normal J-groupmentioning

confidence: 99%

“…The following result is extracted from [3,6,8]: Proposition 2.9. Let S be an elementary normal j-group.…”

Section: Symplectic Symmetric Spaces Always Carry a Preferred Lie Gro...mentioning

confidence: 99%

See 2 more Smart Citations

Deformation Quantization for Actions of Kählerian Lie Groups

Bieliavsky¹,

Gayral

2015

Memoirs of the AMS

Self Cite

135

View full text Add to dashboard Cite

Let B be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action α of B on a Fréchet algebra A. Denote by A ∞ the associated Fréchet algebra of smooth vectors for this action. In the Abelian case B = R 2n and α isometric, Marc Rieffel proved in [26] that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Fréchet algebra structures {⋆ α θ } θ∈R on A ∞ . When A is a C * -algebra, every deformed Fréchet algebra (A ∞ , ⋆ α θ ) admits a compatible pre-C * -structure, hence yielding a deformation theory at the level of C * -algebras too. In this memoir, we prove both analogous statements for general negatively curved Kählerian groups. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geometrical objects coming from invariant WKBquantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderón-Vaillancourt Theorem. In particular, we give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.

show abstract

Section: Star-products On Normal J-groupsmentioning

confidence: 80%

“…where a, t 1 , t 2 , ℓ ∈ R and v 1 , v 2 ∈ V . The group law ofG in these coordinates then reads: 6) and the inversion map is given by:…”

Section: The Transvection Quadruple Of An Elementary Normal J-groupmentioning

confidence: 99%

See 1 more Smart Citation

Deformation Quantization for Actions of Kählerian Lie Groups

Bieliavsky¹,

Gayral

2015

Memoirs of the AMS

Self Cite

135

View full text Add to dashboard Cite

show abstract

“…These considerations therefore lead us to focus on the class of Hermitian symmetric spaces of the non-compact type, which we present here only in the rank one case (the higher rank situation can be treated by applying the 'extension lemma' in [11]). Definition 2.1.…”

Section: Definitions and Notationsmentioning

confidence: 99%

Noncommutative field theory on rank one symmetric spaces

Bieliavsky¹,

Gurău²,

Rivasseau³

2009

J. Noncommut. Geom.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In view of UDF's, we will be concerned with symmetric spaces underlying Lie group manifolds [BCSV07].…”

Section: Symplectic Symmetric Spacesmentioning

confidence: 99%

Deformation quantization for actions of the affine group

Bieliavsky

2007

Preprint

Self Cite

View full text Add to dashboard Cite

We define a universal deformation formula (UDF) for the actions of the affine group on Fréchet algebras. More precisely, starting with any associative Fréchet algebra A which the affine group S ≃ ax+b acts on in a strongly continuous and isometrical manner, the UDF produces a family of topological associative algebra structures on the space A ∞ of smooth vectors of the action deforming the initial product. The deformation field obtained is based over an infinite dimensional parameter space naturally associated with the space of pseudo-differential operators on the real line. This note also presents some geometrical aspects of the UDF and in particular its relation with hyperbolic geometry.

show abstract

Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces

Cited by 7 publications

References 36 publications

Deformation Quantization for Actions of Kählerian Lie Groups

Deformation Quantization for Actions of Kählerian Lie Groups

Noncommutative field theory on rank one symmetric spaces

Deformation quantization for actions of the affine group

Contact Info

Product

Resources

About