Abstract |
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Kac–Moody groups may be viewed as infinite-dimensional analogues of semi-simple Lie groups, or else semi-simple algebraic groups. More precisely, one first considers Kac–Moody algebras, which are — usually infinite-dimensional — complex Lie algebras, and which may be viewed as generalisations of the finite-dimensional semi-simple complex Lie algebras. One can then “exponentiate” such a Kac-Moody algebra to get a group, and even a group functor G over the category of fields, called a Tits functor. A (minimal) Kac–Moody group is then a group of the form G(k), obtained by evaluation of a Tits functor G over a field k. The denotation “minimal” for a Kac–Moody group G(k) is justified by the fact that one can also construct “maximal” versions of G(k), which are obtained from G(k) by completing it with respect to some suitable topology, and thus contain G(k) as a dense subgroup. Minimal Kac–Moody groups G(k) over a field k of characteristic zero are interesting in themselves as infinite-dimensional generalisations of semi-simple real or complex Lie groups. They also seem to appear more and more in connection with theoretical physics. Maximal Kac–Moody groups over finite fields yield a prominent class — one of the very few that are known in fact — of simple, non-linear, totally disconnected and compactly generated locally compact groups. The latter class of groups plays a fundamental role in the structure theory of general (compactly generated) locally compact groups. In this thesis, several results concerning different aspects of the structure of minimal and maximal Kac–Moody groups are established, be it their topological, or infinitesimal, or else abstract group structure. |