Hamiltonian algebroid

Definition 0.1. 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 :

Moreover, consider the classical configuration space Q = ℝ³ of a classical, mechanical system, or particle whose phase space is the cotangent bundle T^∗ℝ³ℝ⁶, 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 ∘ is associative. We recall that one needs to consider first a specific algebra (defined as a vector space E over a ground field (typically ℝ or ℂ)) equipped with a bilinear and distributive multiplication ∘ . Then one defines a Jordan algebra (over ℝ), as a a specific algebra over ℝ for which:

S ∘ T = T ∘ S , S ∘ (T ∘ S²) = (S ∘ T) ∘ S²,,

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 𝔄_ℝ together with a Jordan product ∘ and Poisson bracket

{ , }, satisfying :

• for all S,T ∈ 𝔄_ℝ,

  S ∘ T = T ∘ S {S,T} = −{T,S}

• the Leibniz rule holds

  {S,T ∘ W} = {S,T}∘ W + T ∘{S,W} for all S,T,W ∈ 𝔄_ℝ, along with

• the Jacobi identity :

  

• for some ℏ² ∈ ℝ, 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 L related to such a Poisson structure on the target space.