# 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.
 $\{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}$
• 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