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\begin{center}\textbf{\large On Poisson Geometry}
\par\medskip
\textit{Kirill C.H. Mackenzie}
\end{center}

\bigskip
\noindent
Poisson geometry has three principal sources:

1. It is a more convenient context in which to study
many aspects of symplectic geometry.

2. It is the theory dual to Lie algebra theory and,
more generally, Lie algebroid theory.

3. It is a (semi-)classical limit of quantum theory.

\medskip
Formally, a Poisson structure on a manifold $M$ is a Lie algebra structure on
the real vector space of smooth functions $C(M)$ such that
$$ \{f, gh\} = g\{f,
h\} + h\{f, h\} $$ for all $f, g, h\in C^{\infty}(M)$.

\medskip
If $M$ is a symplectic manifold with symplectic form
$\omega$ then each function $f\in C(M)$ defines the
Hamiltonian vector field $X_f$ by the condition that
$\omega(X_f, Y) = Y(f)$ for all vector fields $Y$. In terms
of these Hamiltonian vector fields, the classical
Poisson bracket
$$
\{f, g\} = -X_f(g)
$$
is a Poisson structure on $M$. That $d\omega = 0$ is precisely
what is needed for the Poisson bracket to satisfy Jacobi.
Conversely any Poisson structure on a manifold $M$ foliates
it by symplectic leaves.

Many constructions in symplectic geometry destroy the
symplectic structure but preserve the Poisson bracket.
Put another way, the only morphisms in the symplectic
category are the symplectomorphisms, which are
diffeomorphisms, but the obvious concept of Poisson
map allows a much greater flexibility and power.

\medskip
If $\mathfrak{g}$ is a Lie algebra then its dual
vector space $M  = \mathfrak{g}^*$ has a natural
Poisson structure. Namely take $f\in C(M)$ and any
point $\phi\in M$. Then the (directional) derivative
of $f$ at $\phi$ is a linear map $M\to \R$; that is,
it is an element of $\mathfrak{g}$. So we can define
$\{f, g\}\in C(M)$ by
$$
\{f, g\}(\phi) = \phi([D(f), D(g)])
$$
where $D(f)$ is the directional derivative of $f$ at
$\phi$. In this example the bracket of two linear functions
is linear and this characterizes duals of Lie algebras
amongst all Poisson structures. There is a lot of evidence
that Lie actually thought more in terms of these duals
than in terms of what we call a Lie algebra.

In this example the symplectic leaves are precisely
the coadjoint orbits of any Lie group which integrates
$\mathfrak{g}$.

If we replace $\mathfrak{g}$ by the tangent bundle to
some manifold $P$ and play the same game with the
bracket of vector fields, we get the (Poisson structure
corresponding to the) standard symplectic structure on
$T^*P$.

\medskip
I won't say much about how Poisson structures are a
first stage in quantization, except that a Poisson bracket
is the first term in the power series of a deformation
quantization. Also, Poisson--Lie groups are very definitely a
good approximation to quantum groups (Poisson groups also
arise in complete integrability) and a good intro can be
found in Chari and Pressley, for example.

\bigskip
I hope this is some help --- and sounds sufficiently seductive.
I will definitely recall the
Marsden--Weinstein reduction at the start of my talk.



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