PlanetPhysics/Hamiltonian Algebroid 2

Introduction
Hamiltonian algebroids are 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{matrix} [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, 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 \bR or \bC)) equipped with a bilinear and distributive multiplication \circ~. Then one defines a Jordan algebra (over \bR), as a a specific algebra over \bR for which:

$$ S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 , ,$$

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 together with a Jordan product  \circ and Poisson bracket

\{~,~\}, satisfying~:

\item[1.] for all  $$ S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\}   \item[2.] the Leibniz rule  holds  $$ \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}$$ for all $$S, T, W \in \mathfrak A_{\bR}$$, along with   \item[3.]  the Jacobi identity~:  $$ \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$$   \item[4.]  for some $$\hslash^2 \in \bR$$, there is the associator identity  ~:  $$(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.