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Locally Compact Quantum Groups. A von Neumann Algebra Approach
In this paper, we give an alternative approach to the theory of locally
compact quantum groups, as developed by Kustermans and Vaes. We develop the
theory completely within the von Neumann algebra framework. At various points,
we also do things differently. We have a different treatment of the antipode.
We obtain the uniqueness of the Haar weights in an early stage. We take
advantage of this fact when deriving the other main results in the theory. We
also give a slightly different approach to duality. Finally, we collect, in a
systematic way, several important formulas. In an appendix, we indicate very
briefly how the -approach and the von Neumann algebra approach eventually
yield the same objects
The Poisson center and polynomial, maximal Poisson commutative subalgebras, especially for nilpotent Lie algebras of dimension at most seven
Let g be a finite dimensional Lie algebra over an algebraically closed field
k of characteristic zero. We collect some general results on the Poisson center
of S(g), including some simple criteria regarding its polynomiality, and also
on certain Poisson commutative subalgebras of S(g). These facts are then used
to finish our earlier work on this subject, i.e. to give an explicit
description for the Poisson center of all indecomposable, nilpotent Lie
algebras of dimension at most seven. Among other things, we also provide a
polynomial, maximal Poisson commutative subalgebra of S(g), enjoying additional
properties. As a by-product we show that a conjecture by Milovanov is valid in
this situation. These results easily carry over to the enveloping algebra U(g).Comment: 48 page
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