Isotropy representation
In differential geometry, the isotropy representation is a natural
to a fixed point.Construction
Given a Lie group action on a manifold M, if Go is the
stabilizer
of a point o (isotropy subgroup at o), then, for each g in Go, fixes o and thus taking the derivative at o gives the map By the chain rule,
and thus there is a representation:
given by
- .
It is called the isotropy representation at o. For example, if is a conjugation action of G on itself, then the isotropy representation at the identity element e is the adjoint representation of .
References
- http://www.math.toronto.edu/karshon/grad/2009-10/2010-01-11.pdf
- https://www.encyclopediaofmath.org/index.php/Isotropy_representation
- Kobayashi, Shoshichi; Nomizu, Katsumi (1996). ISBN 0-471-15733-3.