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 $Q={\mathbb{R}}^{3}$ 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 JordanLie (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 $\u2102$)) 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:
$\begin{array}{cc}\hfill S\circ T& =T\circ S,\hfill \\ \hfill S\circ (T\circ {S}^{2})& =(S\circ T)\circ {S}^{2},\hfill \end{array},$
for all elements $S,T$ of this algebra.
Then, the usual algebraic types of morphisms automorphism^{}, isomorphism^{}, etc.) apply to a JordanLie (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 $S,T\in {U}_{\mathbb{R}},$
$\begin{array}{cc}\hfill S\circ T& =T\circ S\hfill \\ \hfill \{S,T\}& =\{T,S\}\hfill \end{array}$

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 ${\mathrm{\hslash}}^{2}\in \mathbb{R}$, there is the associator^{} identity^{} :
$$(S\circ T)\circ WS\circ (T\circ W)=\frac{1}{4}{\mathrm{\hslash}}^{2}\{\{S,W\},T\}.$$
Thus, the canonical transformations of the Poisson sigma model phase space specified by the JordanLie (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  20130322 18:13:44 
Last modified on  20130322 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 