Isotropy Representation
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Isotropy Representation

In differential geometry, the isotropy representation is a natural linear representation of a Lie group, that is acting on a manifold, on the tangent space 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). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley-Interscience. ISBN 0-471-15733-3.



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