PlanetPhysics/Hamiltonian Algebroid 2

Introduction

Hamiltonian algebroids are generalizations of the [[../BilinearMap/|Lie algebras]] of canonical transformations.

Let $X$ and $Y$ be two [[../NeutrinoRestMass/|vector fields]] on a smooth [[../NoncommutativeGeometry4/|manifold]] $M$, represented here as [[../QuantumOperatorAlgebra4/|operators]] acting on [[../Bijective/|functions]]. Their [[../Commutator/|commutator]], or Lie bracket, $L$, 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 [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]], or [[../Particle/|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) [[../QuantumSpinNetworkFunctor2/|observables]] is taken to be the real [[../NormInducedByInnerProduct/|vector space]] of smooth functions on $M$, and with T being an element of a [[../JordanBanachAndJordanLieAlgebras/|Jordan-Lie (Poisson) algebra]] whose definition is also recalled next. Thus, one defines as in classical [[../NewtonianMechanics/|dynamics]] the [[../PoissonRing/|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 [[../CosmologicalConstant2/|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 $\circ$~. 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 , ,$forallelements<math>S,T$ of this algebra.

Then, the usual [[../CoIntersections/|algebraic]] [[../Bijective/|types]] of [[../TrivialGroupoid/|morphisms]] automorphism, [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]], etc.) apply to a [[../JordanBanachAndJordanLieAlgebras/|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 $\circ$ 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 [[../Cod/|identity]]~:  $\{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}$   \item[4.]  for some Failed to parse (unknown function "\bR"): {\displaystyle \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 [[../JordanBanachAndJordanLieAlgebras/|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.