Английская Википедия:Isotropy representation

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Шаблон:Short description 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 <math>(G, \sigma)</math> 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, <math>\sigma_g: M \to M</math> fixes o and thus taking the derivative at o gives the map <math>(d\sigma_g)_o: T_o M \to T_o M.</math> By the chain rule,

<math>(d \sigma_{gh})_o = d (\sigma_g \circ \sigma_h)_o = (d \sigma_g)_o \circ (d \sigma_h)_o</math>

and thus there is a representation:

<math>\rho: G_o \to \operatorname{GL}(T_o M)</math>

given by

<math>\rho(g) = (d \sigma_g)_o</math>.

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

References

• http://www.math.toronto.edu/karshon/grad/2009-10/2010-01-11.pdf
• https://www.encyclopediaofmath.org/index.php/Isotropy_representation
• Шаблон:Cite book

Шаблон:Differential-geometry-stub