Deformation Quantization for Actions of Kahlerian Lie Groups

Ebook Details

Authors

Pierre Bieliavsky, Victor Gayral

Year 2015
Pages 154
Publisher Amer Mathematical Society
Language en
ISBN 9781470414917
File Size 1.23 MB
File Format PDF
Download Counter 102
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Ebook Description

Let $\mathbb{B}$ be a Lie group admitting a left-invariant negatively curved Kahlerian structure. Consider a strongly continuous action $\alpha$ of $\mathbb{B}$ on a Frechet algebra $\mathcal{A}$. Denote by $\mathcal{A}^\infty$ the associated Frechet algebra of smooth vectors for this action. In the Abelian case $\mathbb{B}=\mathbb{R}^{2n}$ and $\alpha$ isometric, Marc Rieffel proved that Weyl's operator symbol composition formula (the so called Moyal product) yields a deformation through Frechet algebra structures $\{\star_{\theta}^\alpha\}_{\theta\in\mathbb{R}}$ on $\mathcal{A}^\infty$. When $\mathcal{A}$ is a $C^*$-algebra, every deformed Frechet algebra $(\mathcal{A}^\infty,\star^\alpha_\theta)$ admits a compatible pre-$C^*$-structure, hence yielding a deformation theory at the level of $C^*$-algebras too. In this memoir, the authors prove both analogous statements for general negatively curved Kahlerian groups. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geom,etrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderon-Vaillancourt Theorem. In particular, the authors give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.