Hamiltonian algebroids

0.1 Introduction

Hamiltonian algebroids are generalizationsPlanetmathPlanetmath of the Lie algebrasMathworldPlanetmath 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 commutatorMathworldPlanetmathPlanetmath, or Lie bracket, L, is :


Moreover, consider the classical configuration space Q=3 of a classical, mechanical system, or particle whose phase space is the cotangent bundle T*36, 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) algebraMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/JordanBanachAndJordanLieAlgebras) whose definition is also recalled next. Thus, one defines as in classical dynamics the Poisson algebra as a Jordan algebraMathworldPlanetmathPlanetmath in which is associative. We recall that one needs to consider first a specific algebra (defined as a vector spaceMathworldPlanetmath E over a ground field (typically or )) equipped with a bilinearPlanetmathPlanetmath and distributive multiplication  . Then one defines a Jordan algebra (over ), as a a specific algebra over for which:


for all elements S,T of this algebra.

Then, the usual algebraic types of morphisms automorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, etc.) apply to a Jordan-Lie (Poisson) algebra (http://planetmath.org/JordanBanachAndJordanLieAlgebras) defined as a real vector space U together with a Jordan product and Poisson bracket

{,}, satisfying :

  • 1.

    for all S,TU,


  • 2.

    the Leibniz rule holds


    for all S,T,WU, along with

  • 3.
  • 4.

    for some 2, there is the associatorMathworldPlanetmath identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath  :


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 , define a Hamiltonian algebroid with the Lie brackets L related to such a Poisson structureMathworldPlanetmath 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