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The representation of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227401.png" /> contragredient to the adjoint representation Ad of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227402.png" /> (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). The coadjoint representation acts on the dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227403.png" /> of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227404.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227405.png" />.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227406.png" /> is a real matrix group, i.e. a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227407.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227408.png" /> is a subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227409.png" /> of real matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274011.png" /> be the orthogonal complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274012.png" /> relative to the bilinear form
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274013.png" /></td> </tr></table>
+
The representation of a Lie group  $  G $
 +
contragredient to the adjoint representation Ad of  $  G $(
 +
cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). The coadjoint representation acts on the dual  $  \mathfrak g  ^ {*} $
 +
of the Lie algebra  $  \mathfrak g $
 +
of the group  $  G $.
  
let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274014.png" /> be some subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274015.png" /> complementary to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274016.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274017.png" /> be the projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274018.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274019.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274020.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274021.png" /> and the coadjoint representation is given by the formula
+
If  $  G $
 +
is a real matrix group, i.e. a subgroup of  $  \mathop{\rm GL} ( n, \mathbf R ) $,
 +
then  $  \mathfrak g $
 +
is a subspace of the space  $  \mathop{\rm Mat} _ {n} ( \mathbf R ) $
 +
of real matrices of order  $  n $.  
 +
Let  $  \mathfrak g  ^  \perp  $
 +
be the orthogonal complement of  $  \mathfrak g $
 +
relative to the bilinear form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274022.png" /></td> </tr></table>
+
$$
 +
( X, Y)  \rightarrow  \mathop{\rm tr}  XY \ \
 +
\mathop{\rm in} \
 +
\mathop{\rm Mat} _ {n} ( \mathbf R ),
 +
$$
  
The corresponding representation of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274023.png" /> is also called the coadjoint representation. In the case above it is defined by
+
let  $  V $
 +
be some subspace of $  \mathop{\rm Mat} _ {n} ( \mathbf R ) $
 +
complementary to  $  \mathfrak g  ^  \perp  $,
 +
and let  $  P $
 +
be the projection onto  $  V $
 +
parallel to  $  \mathfrak g  ^  \perp  $.  
 +
Then  $  \mathfrak g  ^ {*} $
 +
is identified with  $  V $
 +
and the coadjoint representation is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274024.png" /></td> </tr></table>
+
$$
 +
K ( g) X  = \
 +
P ( gXg  ^ {-} 1 ),\ \
 +
g \in G,\ \
 +
X \in V.
 +
$$
  
The coadjoint representation plays a fundamental role in the [[Orbit method|orbit method]] (see [[#References|[2]]]). Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274025.png" />-orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274026.png" /> in the coadjoint representation carries a canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274027.png" />-invariant [[Symplectic structure|symplectic structure]]. In other words, on each orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274028.png" /> there is a uniquely defined non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274029.png" />-invariant closed differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274030.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274031.png" /> (whence it follows that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274032.png" />-orbits in the coadjoint representation are even-dimensional). To obtain an explicit expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274033.png" /> one proceeds as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274034.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274035.png" /> be the orbit through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274036.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274037.png" /> be tangent vectors to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274038.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274039.png" />. There exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274042.png" /> such that
+
The corresponding representation of the Lie algebra  $  \mathfrak g $
 +
is also called the coadjoint representation. In the case above it is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274043.png" /></td> </tr></table>
+
$$
 +
K ( X) Y  = \
 +
P ( XY - YX),\ \
 +
X \in \mathfrak g ,\ \
 +
Y \in V.
 +
$$
 +
 
 +
The coadjoint representation plays a fundamental role in the [[Orbit method|orbit method]] (see [[#References|[2]]]). Each  $  G $-
 +
orbit  $  \Omega $
 +
in the coadjoint representation carries a canonical  $  G $-
 +
invariant [[Symplectic structure|symplectic structure]]. In other words, on each orbit  $  \Omega $
 +
there is a uniquely defined non-degenerate  $  G $-
 +
invariant closed differential  $  2 $-
 +
form  $  B _  \Omega  $(
 +
whence it follows that all  $  G $-
 +
orbits in the coadjoint representation are even-dimensional). To obtain an explicit expression for  $  B _  \Omega  $
 +
one proceeds as follows. Let  $  F \in \mathfrak g  ^ {*} $,
 +
let  $  \Omega $
 +
be the orbit through the point  $  F $
 +
and let  $  \xi , \eta $
 +
be tangent vectors to  $  \Omega $
 +
at  $  F $.
 +
There exist  $  X $
 +
and  $  Y $
 +
in  $  \mathfrak g $
 +
such that
 +
 
 +
$$
 +
\xi  = K ( X) F,\ \
 +
\eta  = K ( Y) F.
 +
$$
  
 
Then
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274044.png" /></td> </tr></table>
+
$$
 +
B _  \Omega  ( \xi , \eta )  = \
 +
\langle  F, [ X, Y] \rangle.
 +
$$
 +
 
 +
For every  $  X \in \mathfrak g $,
 +
the vector field  $  \xi _ {X} ( F) = K ( X) F $
 +
is Hamiltonian with respect to  $  B _  \Omega  $;  
 +
as its generator (generating function) one can take  $  X $
 +
itself, considered as a linear function on  $  \mathfrak g  ^ {*} $.
  
For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274045.png" />, the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274046.png" /> is Hamiltonian with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274047.png" />; as its generator (generating function) one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274048.png" /> itself, considered as a linear function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274049.png" />.
+
The stabilizer of a point with orbit of maximal dimension in the coadjoint representation is commutative [[#References|[1]]]. The Poisson bracket arising on each orbit generates a single Berezin bracket, which defines the structure of a local Lie algebra (cf. [[Lie algebra, local|Lie algebra, local]]), in the space of smooth functions on $  \mathfrak g  ^ {*} $(
 +
see [[#References|[3]]]). The coordinate expression for the Berezin bracket is
  
The stabilizer of a point with orbit of maximal dimension in the coadjoint representation is commutative [[#References|[1]]]. The Poisson bracket arising on each orbit generates a single Berezin bracket, which defines the structure of a local Lie algebra (cf. [[Lie algebra, local|Lie algebra, local]]), in the space of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274050.png" /> (see [[#References|[3]]]). The coordinate expression for the Berezin bracket is
+
$$
 +
\{ f _ {1} , f _ {2} \}  = \
 +
\sum _ {i, j, k }
 +
c _ {ij}  ^ {k} x _ {k}
 +
\frac{\partial  f _ {1} }{\partial  x _ {i} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274051.png" /></td> </tr></table>
+
\frac{\partial  f _ {2} }{\partial  x _ {j} }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274052.png" /> are the [[structure constant]]s of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274053.png" />.
+
where c _ {ij}  ^ {k} $
 +
are the [[structure constant]]s of $  \mathfrak g $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Bernal,  et al.,  "Représentations des groupes de Lie résolubles" , Dunod  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Kirillov,  "Local Lie algebras"  ''Russian Math. Surveys'' , '''31''' :  4  (1976)  pp. 55–75  ''Uspekhi Mat. Nauk'' , '''31''' :  4  (1976)  pp. 57–76</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Bernal,  et al.,  "Représentations des groupes de Lie résolubles" , Dunod  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Kirillov,  "Local Lie algebras"  ''Russian Math. Surveys'' , '''31''' :  4  (1976)  pp. 55–75  ''Uspekhi Mat. Nauk'' , '''31''' :  4  (1976)  pp. 57–76</TD></TR></table>

Latest revision as of 17:45, 4 June 2020


The representation of a Lie group $ G $ contragredient to the adjoint representation Ad of $ G $( cf. Adjoint representation of a Lie group). The coadjoint representation acts on the dual $ \mathfrak g ^ {*} $ of the Lie algebra $ \mathfrak g $ of the group $ G $.

If $ G $ is a real matrix group, i.e. a subgroup of $ \mathop{\rm GL} ( n, \mathbf R ) $, then $ \mathfrak g $ is a subspace of the space $ \mathop{\rm Mat} _ {n} ( \mathbf R ) $ of real matrices of order $ n $. Let $ \mathfrak g ^ \perp $ be the orthogonal complement of $ \mathfrak g $ relative to the bilinear form

$$ ( X, Y) \rightarrow \mathop{\rm tr} XY \ \ \mathop{\rm in} \ \mathop{\rm Mat} _ {n} ( \mathbf R ), $$

let $ V $ be some subspace of $ \mathop{\rm Mat} _ {n} ( \mathbf R ) $ complementary to $ \mathfrak g ^ \perp $, and let $ P $ be the projection onto $ V $ parallel to $ \mathfrak g ^ \perp $. Then $ \mathfrak g ^ {*} $ is identified with $ V $ and the coadjoint representation is given by the formula

$$ K ( g) X = \ P ( gXg ^ {-} 1 ),\ \ g \in G,\ \ X \in V. $$

The corresponding representation of the Lie algebra $ \mathfrak g $ is also called the coadjoint representation. In the case above it is defined by

$$ K ( X) Y = \ P ( XY - YX),\ \ X \in \mathfrak g ,\ \ Y \in V. $$

The coadjoint representation plays a fundamental role in the orbit method (see [2]). Each $ G $- orbit $ \Omega $ in the coadjoint representation carries a canonical $ G $- invariant symplectic structure. In other words, on each orbit $ \Omega $ there is a uniquely defined non-degenerate $ G $- invariant closed differential $ 2 $- form $ B _ \Omega $( whence it follows that all $ G $- orbits in the coadjoint representation are even-dimensional). To obtain an explicit expression for $ B _ \Omega $ one proceeds as follows. Let $ F \in \mathfrak g ^ {*} $, let $ \Omega $ be the orbit through the point $ F $ and let $ \xi , \eta $ be tangent vectors to $ \Omega $ at $ F $. There exist $ X $ and $ Y $ in $ \mathfrak g $ such that

$$ \xi = K ( X) F,\ \ \eta = K ( Y) F. $$

Then

$$ B _ \Omega ( \xi , \eta ) = \ \langle F, [ X, Y] \rangle. $$

For every $ X \in \mathfrak g $, the vector field $ \xi _ {X} ( F) = K ( X) F $ is Hamiltonian with respect to $ B _ \Omega $; as its generator (generating function) one can take $ X $ itself, considered as a linear function on $ \mathfrak g ^ {*} $.

The stabilizer of a point with orbit of maximal dimension in the coadjoint representation is commutative [1]. The Poisson bracket arising on each orbit generates a single Berezin bracket, which defines the structure of a local Lie algebra (cf. Lie algebra, local), in the space of smooth functions on $ \mathfrak g ^ {*} $( see [3]). The coordinate expression for the Berezin bracket is

$$ \{ f _ {1} , f _ {2} \} = \ \sum _ {i, j, k } c _ {ij} ^ {k} x _ {k} \frac{\partial f _ {1} }{\partial x _ {i} } \frac{\partial f _ {2} }{\partial x _ {j} } , $$

where $ c _ {ij} ^ {k} $ are the structure constants of $ \mathfrak g $.

References

[1] P. Bernal, et al., "Représentations des groupes de Lie résolubles" , Dunod (1972)
[2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[3] A.A. Kirillov, "Local Lie algebras" Russian Math. Surveys , 31 : 4 (1976) pp. 55–75 Uspekhi Mat. Nauk , 31 : 4 (1976) pp. 57–76
How to Cite This Entry:
Coadjoint representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coadjoint_representation&oldid=46372
This article was adapted from an original article by A.A. Kirillov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article