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.
Moreover, consider the classical configuration space of a classical, mechanical system, or particle whose phase space is the cotangent bundle , for which the space of (classical) observables is taken to be the real vector space of smooth functions on , 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 is associative. We recall that one needs to consider first a specific algebra (defined as a vector space over a ground field (typically or )) equipped with a bilinear and distributive multiplication . Then one defines a Jordan algebra (over ), as a a specific algebra over for which:
for all elements 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 together with a Jordan product and Poisson bracket
, satisfying :
the Leibniz rule holds
for all , along with
the Jacobi identity :
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 related to such a Poisson structure on the target space.
|Date of creation||2013-03-22 18:13:44|
|Last modified on||2013-03-22 18:13:44|
|Last modified by||bci1 (20947)|