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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Filtrations and canonical coordinates on nilpotent Lie groups
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by Roe Goodman PDF
Trans. Amer. Math. Soc. 237 (1978), 189-204 Request permission

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.
References
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • 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