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 :

\displaystyle[X,Y](f)=X(Y(f))-Y(X(f)).

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

$\begin{aligned} \displaystyle S\circ T&\displaystyle=T\circ S~{},\\ \displaystyle S\circ(T\circ S^{2})&\displaystyle=(S\circ T)\circ S^{2},\end{% aligned},$

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 (http://planetmath.org/JordanBanachAndJordanLieAlgebras) defined as a real vector space $U_{\mathbb{R}}$ together with a Jordan product $\circ$ and Poisson bracket

$\{~{},~{}\}$ , satisfying :

1.

for all $S,T\in U_{\mathbb{R}},$

$\begin{aligned} \displaystyle S\circ T&\displaystyle=T\circ S\\ \displaystyle\{S,T\}&\displaystyle=-\{T,S\}\end{aligned}$
2.

the Leibniz rule holds

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

for all $S,T,W\in U_{\mathbb{R}}$ , along with
3.

the Jacobi identity :

$\{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}$
4.

for some $\hslash^{2}\in\mathbb{R}$ , 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 (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.

Title	Hamiltonian algebroids
Canonical name	HamiltonianAlgebroids
Date of creation	2013-03-22 18:13:44
Last modified on	2013-03-22 18:13:44
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	42
Author	bci1 (20947)
Entry type	Topic
Classification	msc 81P05
Classification	msc 81R15
Classification	msc 81R10
Classification	msc 81R05
Classification	msc 81R50
Synonym	quantum algebroid
Related topic	HamiltonianOperatorOfAQuantumSystem
Related topic	JordanBanachAndJordanLieAlgebras
Related topic	LieBracket
Related topic	LieAlgebroids
Related topic	QuantumGravityTheories
Related topic	Algebroids
Related topic	RCategory
Related topic	RAlgebroid
Defines	Hamiltonian algebroid
Defines	Jordan algebra
Defines	Poisson algebra