Isomorphisms preserving invariants

Let V and W be finite dimensional real vector spaces and let and be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to and , respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomor...

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Bibliographic Details
Published inGeometriae dedicata Vol. 143; no. 1; pp. 1 - 6
Main Author Schwarz, Gerald W.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.12.2009
Subjects
Algebraic Geometry
Convex and Discrete Geometry
Differential Geometry
Hyperbolic Geometry
Mathematics
Mathematics and Statistics
Original Paper
Projective Geometry
Topology
20G20
4L30
Invariant polynomials
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Summary:Let V and W be finite dimensional real vector spaces and let and be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to and , respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G -orbits to H -orbits and L −1 sends H -orbits to G -orbits, then L induces an isomorphism of Y and Z . Conversely, suppose that f : Y → Z is a germ of a diffeomorphism sending the origin of Y to the origin of Z . Then we show that V and W are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-009-9367-0