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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Filtrations and canonical coordinates on nilpotent Lie groups


Author: Roe Goodman
Journal: Trans. Amer. Math. Soc. 237 (1978), 189-204
MSC: Primary 17B35; Secondary 22E25
DOI: https://doi.org/10.1090/S0002-9947-1978-0469991-X
MathSciNet review: 0469991
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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.


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Article copyright: © Copyright 1978 American Mathematical Society