isotropy representation

{{short description|Linear representation of a group on the tangent space to a fixed point of the group.}}

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 $(G, \sigma)$ on a manifold M, if G_o is the stabilizer of a point o (isotropy subgroup at o), then, for each g in G_o, $\sigma_g: M \to M$ fixes o and thus taking the derivative at o gives the map $(d\sigma_g)_o: T_o M \to T_o M.$ By the chain rule,

: $(d \sigma_{gh})_o = d (\sigma_g \circ \sigma_h)_o = (d \sigma_g)_o \circ (d \sigma_h)_o$

and thus there is a representation:

: $\rho: G_o \to \operatorname{GL}(T_o M)$

given by

: $\rho(g) = (d \sigma_g)_o$ .

It is called the isotropy representation at o. For example, if $\sigma$ is a conjugation action of G on itself, then the isotropy representation $\rho$ at the identity element e is the adjoint representation of $G = G_e$ .

References

http://www.math.toronto.edu/karshon/grad/2009-10/2010-01-11.pdf
https://www.encyclopediaofmath.org/index.php/Isotropy_representation
{{cite book |author1=Kobayashi, Shoshichi |author2=Nomizu, Katsumi | title = Foundations of Differential Geometry, Vol. 1 | publisher=Wiley-Interscience | year=1996|edition=New |isbn = 0-471-15733-3}}

Category:Representation theory of Lie groups