\begin{definition}
\emph{Hamiltonian algebroids} are \emph{generalizations of the Lie algebras of canonical transformations}.
Let $X$ and $Y$ be two vector fields on a smooth manifold $M$, represented here as operators acting on functions.
Their commutator, or Lie bracket, L ,is :
\begin{align*}
[X,Y](f)=X(Y(f))-Y(X(f)).
\end {align*}

Moreover, consider the classical configuration space $Q = \bR^3$ of a classical, mechanical system, or particle whose phase space is the cotangent bundle $T^* \bR^3 \cong \bR^6$, for which the space of (classical)
observables is taken to be the real vector space of smooth functions on $M$, and with T being an element
of a Jordan-Lie (Poisson) algebra whose definition is also recalled next. Thus, one defines as in classical dynamics the \emph{Poisson algebra} as a Jordan algebra in which $\circ$ is associative. We recall that one needs to consider first a \emph{specific algebra} (defined as a vector space $E$ over a ground field (typically $\bR$ or $\bC$)) equipped with a bilinear and distributive multiplication $\circ$~. Then one defines a \emph{Jordan algebra} (over $\bR$), as a a specific algebra over $\bR$ for which:


$ \begin{aligned} S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 ,
\end{aligned}$

for all elements $S, T$ of this algebra.\\

Then, the usual algebraic types of morphisms automorphism, isomorphism, etc.) apply to a Jordan-Lie (Poisson) algebra defined as a real vector space $\mathfrak A_{\bR}$ together with a \emph{Jordan product} $\circ$ and \emph{Poisson bracket}

$\{~,~\}$, satisfying~:

\begin{itemize}
\item[1.] for all $S, T \in \mathfrak A_{\bR},$

$\begin{aligned} S \circ T &= T \circ S \\ \{S, T \} &= - \{T,
S\} \end{aligned}$
$ \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}$
for all $S, T, W \in \mathfrak A_{\bR}$, along with

\item[3.]

the \emph{Jacobi identity}~:\\

$ \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$


\item[4.]

for some $\hslash^2 \in \bR$, there is the \emph{associator identity} ~:
\bigbreak
$(S \circ T) \circ W - S \circ (T \circ W) = \frac{1}{4} \hslash^2 \{\{S, W \}, T \}~.$


\end{itemize}

Thus, the canonical transformations of the Poisson sigma model phase space specified by the Jordan-Lie (Poisson) algebra  (also a Poisson algebra), which is determined by both the Poisson bracket and the \emph{Jordan product} $\circ$, define a \emph{Hamiltonian algebroid} with the Lie brackets $L$ related to such a Poisson structure on the target space.
\end{definition}