A Poincaré-Birkhoff-Witt theorem for infinite dimensional Lie algebras
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 \({\mathfrak a}\).
deformation quantization of the Poisson algebra
scientific article; zbMATH DE number 650886