coadjoint representation について

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coadjoint representation ：ウィキペディア英語版

coadjoint representation
In mathematics, the coadjoint representation

K

G

\mathfrak

denotes the Lie algebra of

G

, the corresponding action of

G

\mathfrak^*

\mathfrak

, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on

G

.
The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups

G

a basic role in their representation theory is played by coadjoint orbit.
In the Kirillov method of orbits, representations of

G

are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of

G

, which again may be complicated, while the orbits are relatively tractable.
==Formal definition==
Let

G

be a Lie group and

\mathfrak

be its Lie algebra. Let

\mathrm : G \rightarrow \mathrm(\mathfrak)

G

. Then the coadjoint representation

K : G \rightarrow \mathrm(\mathfrak^*)

is defined as

\mathrm^*(g^) := \mathrm(g^)^*

. More explicitly,
:

\langle K(g)F, Y \rangle = \langle F, \mathrm(g^)Y \rangle

for

g \in G, Y \in \mathfrak, F \in \mathfrak^*,

where

\langle F, Y \rangle

denotes the value of a linear functional

F

on a vector

Y

.
Let

K_

denote the representation of the Lie algebra

\mathfrak

\mathfrak^*

induced by the coadjoint representation of the Lie group

G

. Then for

X \in \mathfrak, K_(X) = -\mathrm(X)^*

where

\mathrm

\mathfrak

. One may make this observation from the infinitesimal version of the defining equation for

K

above, which is as follows :
:

\langle K_(X)F, Y \rangle = \langle F, - \mathrm(X)Y \rangle

for

X, Y \in \mathfrak, F \in \mathfrak^*

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