Filtrations and canonical coordinates on nilpotent Lie groups

Abstract:

Let $\mathfrak {g}$ be a finite-dimensional nilpotent Lie algebra over a field of characteristic zero. Introducing the notion of a positive, decreasing filtration $\mathcal {F}$ on $\mathfrak {g}$, the paper studies the multiplicative structure of the universal enveloping algebra $U(\mathfrak {g})$, and also transformation laws between $\mathcal {F}$-canonical coordinates of the first and second kind associated with the Campbell-Hausdorff group structure on $\mathfrak {g}$. The basic technique is to exploit the duality between $U(\mathfrak {g})$ and $S({\mathfrak {g}^\ast })$, the symmetric algebra of ${\mathfrak {g}^\ast }$, making use of the filtration $\mathcal {F}$. When the field is the complex numbers, the preceding results, together with the Cauchy estimates, are used to obtain estimates for the structure constants for $U(\mathfrak {g})$. These estimates are applied to construct a family of completions $U{(\mathfrak {g})_\mathfrak {M}}$ of $U(\mathfrak {g})$, on which the corresponding simplyconnected Lie group G acts by an extension of the adjoint representation.