A universal deformation formula for \(\mathcal H_{1}\) without projectivity assumption
type: Item
instance of: scholarly article
title: A universal deformation formula for \(\mathcal H_{1}\) without projectivity assumption
zbMATH Open document ID: 1166.58006
DOI: 10.4171/JNCG/34
author: Xiang Tang
author: Yi-Jun Yao
published in: Journal of Noncommutative Geometry
publication date: 2009-05-05T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: The authors aim to find an \textit{universal deformation formula} for Connes-Moscovici's Hopf subalgebra \({\mathcal H}^1_{CM}\subset {\mathcal H}_{CM}\), by using a Fedosov's quantization type for symplectic diffeomorphisms. By universal deformation formula of a Hopf algebra \(A\), they mean an element \(R\in A\hbar\bigotimes_{{\mathbb C}\hbar} A\hbar\) satisfying the following conditions: (a) \(((\triangle\otimes 1)R)(R\otimes 1)=((1\otimes\triangle)R)(1\otimes R)\); (b) \((\epsilon\otimes 1)(R)=1\otimes 1=(1\otimes\epsilon)(R)\). Here \(\triangle\) is the coproduct and \(\epsilon\) the counit in \(A\). The Connes-Moscovici Hopf algebra \({\mathcal H}_{CM}\) is the enveloping algebra of the Lie algebra which is the linear space \(<X, Y,\delta_n>_{n\geq 1}\), with the following brackets: \[ (\clubsuit)\quad[X,Y]=X,\quad [Y,\delta_n]=n\delta_n,\quad[\delta_n,\delta_m]=0, \quad[X,\delta_n]=\delta_{n+1},\quad m,n\geq 1. \] The coproduct \(\triangle\) in \({\mathcal H}_{CM}\) is defined by: \(\triangle(Y)=Y\otimes 1+1\otimes Y\), \(\triangle(X)=X\otimes 1+1\otimes X+\delta_1\otimes Y\), \(\triangle(\delta_1)=\delta_1\otimes 1+1\otimes \delta_1\), where \(\triangle(\delta_n)\) is defined recursively, using above brackets \((\clubsuit)\) and the identity \(\triangle(h_1h_2)=\triangle(h_1)\triangle(h_2)\), \(\forall h_1,h_2\in {\mathcal H}_{CM}\). The algebra generated by \(\{\delta_n\}_{n\geq 1}\) is the graded sub-Hopf algebra \({\mathcal H}^1_{CM}\subset {\mathcal H}_{CM}\), where the gradiation is defined by \(gr(\delta_{n_1},\dots,\delta_{n_s})=n_1+\dots+n_s\). \({\mathcal H}^1_{CM}\) is Hopf-isomorphic to the so-called \textit{de Fadd di Bruno algebra}, \({\mathcal H}_{FB}\equiv{\mathbb R}[a_1,\dots,a_n,\dots]\), where \(a_n:G\to{\mathbb R}\) are suitable functions over the group \(G\) of so-called \textit{formal tangent diffeomorphisms}. The \textit{Fedosov's quantization} of a symplectic manifold \((M,\omega)\) is obtained by a bijection \(f\mapsto\hat…
reviewed by: Agostino Prástaro
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 58H15
Mathematics Subject Classification ID: 53D50
zbMATH DE Number: 5550886
zbMATH Keywords: Connes-Moscovici Hopf algebras
zbMATH Keywords: groupoid
zbMATH Keywords: deformation
MaRDI profile type: Publication
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OpenAlex ID: W2058259523
description: scientific article; zbMATH DE number 5550886