Nonlocality of equivariant star products on \(T^*(\mathbb{R}\mathbb{P}^n)\)
The author considers the \(\text{SL}_{n+1}(\mathbb R)\)-invariant star products \(\star_\lambda\) on the smooth functions on \(T^*( \mathbb R \mathbb P^n)\) that are polynomials in the fibre variables which are constructed using the \(\text{SL}_{n+1} (\mathbb R)\)-equivariant symbol calculus defined in [\textit{P. B. A. Lecomte}, and \textit{V. Yu. Ovsienko}, Lett. Math. Phys. 49, 173-196 (1999; Zbl 0989.17015)]. This symbol calculus assigns to every polynomial function \(\phi\) on \(T^*( \mathbb R \mathbb P^n)\) a polynomial \({\mathcal Q}_{\lambda,t}(\phi)\) in the formal parameter \(t\) with values in the differential operators on the space of smooth \(\lambda\)-densities on \(\mathbb R \mathbb P^n\) and the product \(\star_\lambda\) is defined by pulling back the composition of such differential operators with the mapping \({\mathcal Q}_{\lambda,t}\). The aim of the present paper is to show that the so-defined star products are only well-defined on the functions that are polynomials in the fibre variables and cannot be extended to \(C^\infty( T^*( \mathbb R \mathbb P^n))\) and hence do not define star products in the usual sense [cf. \textit{F. Bayen}, \textit{M. Flato}, \textit{C. Frønsdal}, \textit{A. Lichnerowicz} and \textit{D. Sternheimer}, Ann. Phys. 111, 61-151, 111-151 (1978; Zbl 0377.53024 and Zbl 0377.53025)]. It is shown that the reason for this phenomenon is the occurrence of non-local combinations of the fibrewise Euler vector field \(E\) that are only well-defined on polynomials along the fibres. A similar observation of the fact that there are symbolic calculi on cotangent bundles that only yield associative products on the polynomials along the fibres that are not extensible to star products on all smooth functions has also been made in a very simple example on \(T^*\mathbb R\) in [\textit{M. Bordemann}, \textit{N. Neumaier} and \textit{S. Waldmann}, Commun. Math. Phys. 198, 363-396 (1998; Zbl 0968.53056)]. In order to illustrate this nonlocality the author explicitly computes the cochains \(C_p\) describing \(\star_{1/2}\) in case \(n\) is odd and \(p \geq 1\) -- these cochains are expressed as combinations of bidifferential operators and operators that contain terms of the shape \((E + z/2)^{-1}\) with odd \(z \in \mathbb Z\) that are only defined on the polynomials along the fibres -- and finally gives a concrete formula for the operator \(\psi \mapsto C_p(\phi,\psi)\), where \(\phi\) and \(\psi\) are polynomial functions on \(T^*(\mathbb R \mathbb P^n)\) and \(\phi\) is additionally assumed to be homogeneous.
\(SL_{n+1}(\mathbb{R})\)-equivariant symbol calculus
scientific article; zbMATH DE number 1679299