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This article is cited in 7 scientific papers (total in 7 papers)
Conformally equivariant quantization – a complete classification
Jean-Philippe Michel University of Luxembourg, Campus Kirchberg, Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg City, Luxembourg
Abstract:
Conformally equivariant quantization is a peculiar map between symbols of real weight $\delta$ and differential operators acting on tensor densities, whose real weights are designed by $\lambda$ and $\lambda+\delta$. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight $\delta$. Later, Silhan has determined the critical values of $\delta$ for which unique existence is lost, and conjectured that for those values of $\delta$ existence is lost for a generic weight $\lambda$. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of $\delta$ and $\lambda$. Namely, (i) unique existence is lost if and only if there is a nontrivial conformally invariant differential operator on the space of symbols of weight $\delta$, and (ii) in that case the conformally equivariant quantization exists only for a finite number of $\lambda$, corresponding to nontrivial conformally invariant differential operators on $\lambda$-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization.
Keywords:
quantization, (bi-)differential operators, conformal invariance, Lie algebra cohomology.
Received: July 29, 2011; in final form April 11, 2012; Published online April 15, 2012
Citation:
Jean-Philippe Michel, “Conformally equivariant quantization – a complete classification”, SIGMA, 8 (2012), 022, 20 pp.
Linking options:
https://www.mathnet.ru/eng/sigma699 https://www.mathnet.ru/eng/sigma/v8/p22
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