Hamiltonian algebroids

0.1 Introduction

Hamiltonian algebroids are generalizations of the Lie algebras of canonical transformations, but cannot be considered just a special case of Lie algebroids. They are instead a special case of a http://planetphysics.org/encyclopedia/QuantumAlgebroid.htmlquantum 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 :

$[X,Y](f)=X(Y(f))-Y(X(f)).$

Moreover, consider the classical configuration space $Q={ℝ}^3$ of a classical, mechanical system, or particle whose phase space is the cotangent bundle $T^{*}{ℝ}^3\cong {ℝ}^6$, 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 elementof a Jordan-Lie (Poisson) algebra (http://planetmath.org/JordanBanachAndJordanLieAlgebras) 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 $ℝ$ or $ℂ$)) equipped with a bilinear and distributive multiplication $\circ$ . Then one defines a Jordan algebra (over $ℝ$), as a a specific algebra over $ℝ$ for which:

for all elements $S,T$ of this algebra.

Then, the usual algebraic types of morphisms automorphism, isomorphism, etc.) apply to aJordan-Lie (Poisson) algebra (http://planetmath.org/JordanBanachAndJordanLieAlgebras) defined as a real vector space $U_{ℝ}$ together with a Jordan product $\circ$ and Poisson bracket

$\{,\}$, satisfying :

• for all $S,T\in U_{ℝ},$

• the Leibniz rule holds

  $$\{S,T\circ W\}=\{S,T\}\circ W+T\circ \{S,W\}$$

  for all $S,T,W\in U_{ℝ}$, along with

• the Jacobi identity :

  $$\{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}$$

• for some ${\mathrm{\hslash }}^2\in ℝ$, there is the associator identity  :

  $$(S\circ T)\circ W-S\circ (T\circ W)=\frac{1}{4}{\mathrm{\hslash }}^2\{\{S,W\},T\}.$$

Thus, the canonical transformations of the Poisson sigma model phase space specified by the Jordan-Lie (Poisson) algebra (http://planetmath.org/JordanBanachAndJordanLieAlgebras) (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.