Deformation Quantization for Actions of Kählerian Lie Groups
(2015) Memoirs of the American Mathematical Society — Vol. 236, n° 1115 (2015)
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- <p> Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper B"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper B"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a Fréchet algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Denote by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {A}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the associated Fréchet algebra of smooth vectors for this action. In the Abelian case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper B equals double-struck upper R Superscript 2 n"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">B</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {B}=\mathbb {R}^{2n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> isometric, Marc Rieffel proved that Weyl’s operator symbol composition formula (the so called Moyal product) yields a deformation through Fréchet algebra structures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace star Subscript theta Superscript alpha Baseline right-brace Subscript theta element-of double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msubsup> <mml:mo> ⋆ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> θ </mml:mi> </mml:mrow> <mml:mi> α </mml:mi> </mml:msubsup> <mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> θ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\{\star _{\theta }^\alpha \}_{\theta \in \mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {A}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra, every deformed Fréchet algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper A Superscript normal infinity Baseline comma star Subscript theta Superscript alpha Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mo> ⋆ </mml:mo> <mml:mi> θ </mml:mi> <mml:mi> α </mml:mi> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathcal {A}^\infty ,\star ^\alpha _\theta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a compatible pre- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -structure, hence yielding a deformation theory at the level of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 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 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. </p>
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Citations
Bieliavsky, P., & Gayral, V. (2015). Deformation Quantization for Actions of Kählerian Lie Groups. Memoirs of the American Mathematical Society, 236(1115). https://doi.org/10.1090/memo/1115 (Original work published 2015)