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 all Failed 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 all Failed to parse (unknown function "\bR"): {\displaystyle S, T, W \in \mathfrak A_{\bR}}
, along with   \item[3.]  the Jacobi identity~:      \item[4.]  for some Failed to parse (unknown function "\bR"): {\displaystyle \hslash^2 \in \bR}
, there is the associator 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.