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Some norms on universal enveloping algebras. (English) Zbl 0908.17007

Let \({\mathfrak g}\) be a (finite-dimensional) complex Lie algebra, \(U\) the universal enveloping algebra of \(g\) and \(U'\) the algebraic dual of \(U\). The first result states that \(U'\) is generated by finite rank elements if and only if \({\mathfrak g}\) is nilpotent. The author presents a kind of dual to the PBW theorem and with its help computes the Taylor coefficients of some functions in the case when \({\mathfrak g}\) is the Lie algebra of a Lie group \(G\). Finally, two kind of norms (seminorms) on \(U\) and on subspaces of \(U'\) are discussed.
The main result here states that the natural isomorphism between the universal enveloping algebra of a direct sum of real (complex) Lie algebras and the tensor product of the corresponding universal enveloping algebras is isometric with respect to certain natural norms on the universal enveloping algebras.

MSC:

17B35 Universal enveloping (super)algebras
16S30 Universal enveloping algebras of Lie algebras
22E60 Lie algebras of Lie groups
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