PlanetPhysics/Hamiltonian Algebroid 2

Introduction

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

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 :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 $M$, 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 $\circ$ is associative. We recall that one needs to consider first a specific algebra (defined as a vector space $E$ over a ground field (typically Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 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 $\circ$ 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle S, T, W \in \mathfrak A_{\bR}}
, along with   \item[3.]  the Jacobi 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 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.