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 essenc ...

of a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

, that is

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

on a manifold, on the

tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

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 and first properties Let \sigma: G \times M \to M, ( ...

(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 In calculus, the chain rule is a formula that expresses the derivative of the 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(x)=f(g(x)) for every , ...

, :

(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 chang ...

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 G ...

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