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Some Norms on Universal Enveloping Algebras

Published online by Cambridge University Press:  20 November 2018

Leonard Gross
Leonard Gross*
Affiliation:
Department of Mathematics Cornell University Ithaca, New York 14853 U.S.A., e-mail: [email protected]
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Abstract

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The universal enveloping algebra, $U(\mathfrak{g})$, of a Lie algebra $\mathfrak{g}$ supports some norms and seminorms that have arisen naturally in the context of heat kernel analysis on Lie groups. These norms and seminorms are investigated here from an algebraic viewpoint. It is shown that the norms corresponding to heat kernels on the associated Lie groups decompose as product norms under the natural isomorphism $U({{\mathfrak{g}}_{1\,}}\oplus \,{{\mathfrak{g}}_{2}})\,\cong \,U({{\mathfrak{g}}_{1}})\,\otimes \,U({{\mathfrak{g}}_{2}})$. The seminorms corresponding to Green's functions are examined at a purely Lie algebra level for $\text{sl(2,}\,\mathbb{C}\text{)}$. It is also shown that the algebraic dual space ${U}'$ is spanned by its finite rank elements if and only if $\mathfrak{g}$ is nilpotent.

Keywords

17B3516S3022E30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 2 , 01 April 1998 , pp. 356 - 377
Copyright
Copyright © Canadian Mathematical Society 1998

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