JHEP07(2019)045
g × g
∗
→ k, by
hµ, ξi ≡ µ(ξ) , (2.1)
where µ ∈ g
∗
and ξ ∈ g.
Define a Poisson bracket on g
∗
, called the Lie-Poisson bracket, by
{F, G}(µ) ≡
µ,
δF
δµ
,
δG
δµ
. (2.2)
F and G are functions on g
∗
and their functional derivatives are considered as elements of
g. The bracket [·, ·] is the Lie bracket on g. The Lie-Poisson bracket (2.2) is clearly bilinear
and antisymmetric, and it can be checked that it is a derivation and satisfies the Jacobi
identity. This makes g
∗
a Poisson manifold. A Hamiltonian vector field, X
H
, is defined in
the usual way: it is a vector field on g
∗
for which there exists a function, H, such that
X
H
[G] = {G, H}, (2.3)
for all functions G. Flows of Hamiltonian vector fields on g
∗
are classical dynamical systems.
The Lie-Poisson bracket defines a foliation of g
∗
into a family of symplectic leaves. Two
points in g
∗
are said to be on the same symplectic leaf if there is a piecewise smooth curve
in g
∗
joining the points such that each segment of the curve is a locally defined Hamiltonian
vector field. The coadjoint orbits of G can be succinctly described as the symplectic leaves
of the Lie-Poisson bracket on g
∗
. The restriction of the Lie-Poisson bracket (2.2) to a
coadjoint orbit is symplectic.
The reason for the name coadjoint orbit is as follows. The adjoint action of G on g by
conjugation is
g · ξ = gξg
−1
, (2.4)
with an infinitesimal version
ξ · ξ
0
= [ξ, ξ
0
] . (2.5)
This defines a map from G to the automorphism group of g called the adjoint representation.
The coadjoint action of G on g
∗
is defined using the pairing (2.1),
hg · µ, ξi = hµ, g
−1
· ξi, (2.6)
for all ξ ∈ g, with an infinitesimal version
hξ · µ, ξ
0
i = −hµ, [ξ, ξ
0
]i, (2.7)
for all ξ
0
∈ g. This defines a map to the automorphism group of g
∗
called the coadjoint
representation. It turns out that the orbits of the coadjoint action of G on g
∗
coincide with
the symplectic leaves of the Lie-Poisson bracket (see [43]).
At a point, µ, on a coadjoint orbit, we identify tangent vectors,
˜
ξ, with Lie algebra
elements, ξ ∈ g, using ξ · µ =
˜
ξ. The Kirillov-Kostant symplectic form acts on tangent
vectors as
ω
K
(
˜
ξ,
˜
ξ
0
) ≡ hµ, [ξ, ξ
0
]i . (2.8)
It is the symplectic form induced by the restriction of the Lie-Poisson bracket to the orbit.
It plays a key role in all that follows.
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