A Poincaré-Birkhoff-Witt theorem for infinite dimensional Lie algebras
type: Item
instance of: scholarly article
title: A Poincaré-Birkhoff-Witt theorem for infinite dimensional Lie algebras
zbMATH Open document ID: 0803.17007
author: Hideki Omori
author: Yoshiaki Maeda
author: Akira Yoshioka
published in: Journal of the Mathematical Society of Japan
publication date: 1994-10-16T00:00:00Z
review text: Let \({\mathfrak g}\) be an ILH-Lie algebra, i.e., the infinite dimensional Lie algebra with the inverse limit topology of certain Hilbert Lie algebras. Let \({\mathfrak g}^*\) denote the dual space of \({\mathfrak g}\). We put \({\mathfrak a}= C^ \infty ({\mathfrak g}^*)\), the space of all smooth functions on \({\mathfrak g}^*\). On \({\mathfrak a}\), there exists a natural Poisson structure \(\{ , \}\), called Lie-Poisson structure, which is given as \(\{f,g\} (p)= [dg(p), dg(p)]'\), \(p\in{\mathfrak g}^*\), \(\forall f,g\in{\mathfrak a}\), where \([ , ]'\) is the Lie bracket of \({\mathfrak g}\). Namely \(({\mathfrak a}, \{ , \})\) is a Poisson algebra. We consider a deformation quantization of the Poisson algebra \(({\mathfrak a},\{ , \})\) in the sense of \textit{F. Bayen}, \textit{M. Flato}, \textit{C. Fronsdal}, \textit{A. Lichnerowicz} and \textit{D. Sternheimer} [Ann. Phys. 111, 61-110 (1978; Zbl 0377.53024)]. In this paper, we show that \(({\mathfrak a},\{ ,\})\) is deformation quantizable. The obtained noncommutative algebra has the property that \(\pi_ m ({\mathfrak g},{\mathfrak g})=0\) for \(m\geq 2\), where the star product is written in the form \[ f*g= fg- {\textstyle {\nu\over 2}} \{f,g\}+ \nu^ 2 \pi_ 2 (f,g)+ \cdots+ \nu^ m \pi_ m (f,g)+\cdots, \qquad f,g\in{\mathfrak a}. \] The quantized algebra naturally contains the universal enveloping algebra of the Lie algebra generated by \({\mathfrak g}\) and \(\nu\) with the relations \([X,Y]= \nu[ X,Y]'\), \(X,Y\in{\mathfrak g}\). Here \([ , ]\) is the Lie bracket induced from * and \(X,Y\in {\mathfrak g}\) is viewed as elements of \({\mathfrak a}\).
reviewed by: Akira Yoshioka
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 650886
zbMATH Keywords: infinite dimensional Lie algebra
zbMATH Keywords: inverse limit topology
zbMATH Keywords: Hilbert Lie algebras
zbMATH Keywords: deformation quantization of the Poisson algebra
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04610025
OpenAlex ID: W1968674819
DOI: 10.2969/JMSJ/04610025
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description: scientific article; zbMATH DE number 650886