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The structure of (various dense subgroups of) the group of automorphisms of a circle

Algebraic Geometry Seminar

The structure of (various dense subgroups of) the group of automorphisms of a circle
Series: Algebraic Geometry Seminar
Location: ENR2 S395
Presenter: Douglas Pickrell, University of Arizona

The group of automorphisms of a compact manifold is a (infinite dimensional) Lie group. In the simplest case of a circle (i.e. closed string), there are well-known structural similarities with the group of automorphisms of a vector space. In the latter case, it is known how to express the group in terms of generating (root) subgroups and relations (basically due to Steinberg, corresponding to the Serre relations for the Lie algebra); these relations are aptly described by Kac as analytic continuations of the relations for the symmetric group (the Weyl group). The point of the talk is to explain that for (various skeletal) subgroups of the group of automorphisms of a circle, the relations disappear (in terms of heuristics, because the Weyl group is trivial). This involves some rudimentary algebraic geometry.

Prerequisites: knowledge of Lie theory would be helpful (to understand the motivation), but knowledge of group theory is all that is needed for understanding the basic results.