Tangential deformations on the dual of nilpotent special Lie algebras
The relationship between \(*\)-products (formal deformations) and quantization deformations (non formal deformations) on dual of nilpotent Lie algebras are studied. An explicit, tangential quantization deformation is given on algebras of polynomial functions and \(C^\infty\), rapidly decreasing functions on the dual of any nilpotent special Lie algebras. This paper is organized as follows. In the first section, we recall the definition of \(*\)-product, we show that Rieffel's deformation quantization can be viewed as a \(*\)-product, we extend it to the space \({\mathcal P} ({\mathfrak g}^*)\) of polynomial functions on \({\mathfrak g}^*\) and give an explicit expression of it. In the second section, we construct a convergent tangential and graded \(*\)-product on \({\mathcal S}({\mathfrak g}^*) \oplus {\mathcal P} ({\mathfrak g}^*)\) in the case of a special nilpotent Lie algebra \({\mathfrak g}\). This product coincides on \({\mathcal P}({\mathfrak g}^*)\) with the formal product of [\textit{D. Arnal}, \textit{M. Cahen} and \textit{S. Gutt}, J. Geom. Phys. 3, 327-351 (1986; Zbl 0616.58013)], but the expressions we give here are totally explicit and have integral form which is much more adapted for use. In the third section, we define the \(*\)-exponential map and the adapted Fourier transform and we relate it to unitary representations of the corresponding Lie group. Finally, we compare our construction to those of Arnal, Cahen, Gutt and of Howe, Ratcliff, Wildberger.
scientific article; zbMATH DE number 910230