Coadjoint representation
The representation of a Lie group contragredient to the adjoint representation Ad of
(cf. Adjoint representation of a Lie group). The coadjoint representation acts on the dual
of the Lie algebra
of the group
.
If is a real matrix group, i.e. a subgroup of
, then
is a subspace of the space
of real matrices of order
. Let
be the orthogonal complement of
relative to the bilinear form
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let be some subspace of
complementary to
, and let
be the projection onto
parallel to
. Then
is identified with
and the coadjoint representation is given by the formula
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The corresponding representation of the Lie algebra is also called the coadjoint representation. In the case above it is defined by
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The coadjoint representation plays a fundamental role in the orbit method (see [2]). Each -orbit
in the coadjoint representation carries a canonical
-invariant symplectic structure. In other words, on each orbit
there is a uniquely defined non-degenerate
-invariant closed differential
-form
(whence it follows that all
-orbits in the coadjoint representation are even-dimensional). To obtain an explicit expression for
one proceeds as follows. Let
, let
be the orbit through the point
and let
be tangent vectors to
at
. There exist
and
in
such that
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Then
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For every , the vector field
is Hamiltonian with respect to
; as its generator (generating function) one can take
itself, considered as a linear function on
.
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 (see [3]). The coordinate expression for the Berezin bracket is
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where are the structure constants of
.
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 |
Coadjoint representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coadjoint_representation&oldid=18625