# PlanetPhysics/Hamiltonian Algebroid 2

### IntroductionEdit

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

Let and be two vector fields on a smooth manifold , represented here as operators acting on functions. Their commutator, or Lie bracket, , is :

**Failed to parse (unknown function "\begin{matrix}"): {\displaystyle \begin{matrix} [X,Y](f)=X(Y(f))-Y(X(f)). \end {align*} Moreover, consider the classical configuration space <math>Q = \bR^3}**
of a classical, mechanical system, or particle whose phase space is the cotangent bundle **Failed to parse (unknown function "\bR"): {\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 , 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 is associative. We recall that one needs to consider first a *specific algebra* (defined as a vector space over a ground field (typically **Failed to parse (unknown function "\bR"): {\displaystyle \bR}**
or **Failed to parse (unknown function "\bC"): {\displaystyle \bC}**
)) equipped with a bilinear and distributive multiplication ~. Then one defines a *Jordan algebra* (over **Failed to parse (unknown function "\bR"): {\displaystyle \bR}**
), as a a specific algebra over **Failed to parse (unknown function "\bR"): {\displaystyle \bR}**
for which:

</math> S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 , , 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 **Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}}**
together with a *Jordan product* and *Poisson bracket*

, satisfying~:

\item[1.] for allFailed to parse (unknown function "\bR"): {\displaystyle S, T \in \mathfrak A_{\bR},}</math> S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\}Failed to parse (unknown function "\item"): {\displaystyle \item[2.] the ''Leibniz rule'' holds <math> \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}}for allFailed to parse (unknown function "\bR"): {\displaystyle S, T, W \in \mathfrak A_{\bR}}, along with \item[3.] theJacobi identity~: \item[4.] for someFailed to parse (unknown function "\bR"): {\displaystyle \hslash^2 \in \bR}, there is theassociator identity~:

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* , define a *Hamiltonian algebroid* with the Lie brackets related to such a Poisson structure on the target space.