# PlanetPhysics/Hamiltonian Algebroid 2

### Introduction

Hamiltonian algebroids are generalizations of the Lie algebras of canonical transformations.

Let ${\displaystyle X}$  and ${\displaystyle Y}$  be two vector fields on a smooth manifold ${\displaystyle M}$ , represented here as operators acting on functions. Their commutator, or Lie bracket, ${\displaystyle L}$ , is :

matrix}"): {\displaystyle \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 $\displaystyle T^* \bR^3 \cong \bR^6$ , for which the space of (classical) observables is taken to be the real vector space of smooth functions on ${\displaystyle M}$ , 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 ${\displaystyle \circ }$ is associative. We recall that one needs to consider first a specific algebra (defined as a vector space ${\displaystyle E}$ over a ground field (typically $\displaystyle \bR$ or $\displaystyle \bC$ )) equipped with a bilinear and distributive multiplication ${\displaystyle \circ }$ ~. Then one defines a Jordan algebra (over $\displaystyle \bR$ ), as a a specific algebra over $\displaystyle \bR$ for which:$ S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 , ,${\displaystyle forallelements$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 $\displaystyle \mathfrak A_{\bR}$ together with a Jordan product ${\displaystyle \circ }$ and Poisson bracket ${\displaystyle \{~,~\}}$ , satisfying~:  \item[1.] for all $\displaystyle S, T \in \mathfrak A_{\bR},$$ S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\} $\displaystyle \item[2.] the ''Leibniz rule'' holds [itex] \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}$
for all $\displaystyle S, T, W \in \mathfrak A_{\bR}$
, along with   \item[3.]  the Jacobi identity~:  ${\displaystyle \{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}}$    \item[4.]  for some $\displaystyle \hslash^2 \in \bR$
, 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 ${\displaystyle \circ }$ , define a Hamiltonian algebroid with the Lie brackets ${\displaystyle L}$  related to such a Poisson structure on the target space.