Algebroids

Definition 0.1 Let $X$

and

be two vector fields on a smooth manifold $M$

, represented here as operators acting on functions. Their commutator, or Lie bracket, $L$

, is :

$\displaystyle [X,Y](f)=X(Y(f))-Y(X(f)).$

Moreover, consider the classical configuration space $Q = \mathbb{R}^3$ of a classical, mechanical system, or particle whose phase space is the cotangent bundle $T^* \mathbb{R}^3 \cong \mathbb{R}^6$ , 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 Poisson algebra as a Jordan algebra in which $\circ$ is associative. We recall that one needs to consider first a specific algebra (defined as a vector space $E$ over a ground field (typically $\mathbb{R}$ or $\mathbb{C}$ )) equipped with a bilinear and distributive multiplication $\circ$ . Then one defines a Jordan algebra (over $\mathbb{R}$ ), as a a specific algebra over $\mathbb{R}$ 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 $\mathfrak{A}_{\mathbb{R}}$ together with a Jordan product $\circ$ and Poisson bracket

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

1.	for all $S, T \in \mathfrak{A}_{\mathbb{R}},$ $\begin{equation}\begin{aligned}S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\} \end{aligned}\end{equation}$
2.	the Leibniz rule holds $\{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}$ for all $S, T, W \in \mathfrak{A}_{\mathbb{R}}$ , along with
3.	the Jacobi identity : $\displaystyle \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$
4.	for some $\hslash^2 \in \mathbb{R}$ , there is the associator identity : $\displaystyle (S \circ T) \circ W - S \circ (T \circ W) = \frac{1}{4} \hslash^2 \{\{S, W \}, T \}~.$

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

Introduction