In differential geometry, the isotropy representation is a natural

linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

of a

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

, that is

acting Acting is an activity in which a story is told by means of its enactment by an actor who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad range of sk ...

on a manifold, on the

tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...

to a fixed point.

Construction

Given a

Lie group action In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable. __TOC__ Definition Let \sigma: G \times M \to M, (g, x) \mapsto g \cdot ...

(G, \sigma)

on a manifold ''M'', if ''G''_''o'' is the

stabilizer Stabilizer, stabiliser, stabilisation or stabilization may refer to: Chemistry and food processing * Stabilizer (chemistry), a substance added to prevent unwanted change in state of another substance ** Polymer stabilizers are stabilizers used ...

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 In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...

, :

(d \sigma_)_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(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 Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change o ...

action of ''G'' on itself, then the isotropy representation

\rho

at the identity element ''e'' is the

adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is \m ...

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 * {{differential-geometry-stub Representation theory of Lie groups