Kontsevich quantization and invariant distributions on Lie groups

1999/10/20 by Martin Andler, Andler, Martin, Alexander Dvorsky +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.DG #math.QA #math.RT

paper · pdf · doi:10.48550/arxiv.math/9910104

22 pages, LaTeX. This is an expanded version of math.QA/9905065

arxiv created 1999/10/20 · arxiv updated 2009/11/30

Abstract

We study Kontsevich's deformation quantization for the dual of a finite-dimensional real Lie algebra (or superalgebra) g. In this case the Kontsevich star-product defines a new convolution on S(g), regarded as the space of distributions supported at 0 in g. For p in S(g), we show that the convolution operator f->f*p is a differential operator with analytic germ. We use this fact to prove a conjecture of Kashiwara and Vergne on invariant distributions on a Lie group. This yields a new proof of Duflo's result on local solvability of bi-invariant differential operators on a Lie group. Moreover, this new proof extends to Lie supergroups.

Related

• Convolution of invariant distributions: Proof of the Kashiwara-Vergne conjecture
• Deformation Quantisation via Kontsevich Formality Theorem
• Variations on deformation quantization
• About the convolution of distributions on groupoids
• Deformation Quantization: Genesis, Developments and Metamorphoses