Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Molecular Hamiltonian wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Bra–ket notation wikipedia , lookup
Two-dimensional conformal field theory wikipedia , lookup
Vertex operator algebra wikipedia , lookup
Dirac bracket wikipedia , lookup
Lie algebra extension wikipedia , lookup
Hamiltonian algebroids∗ bci1† 2013-03-22 1:22:49 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 quantum 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 = R3 of a classical, mechanical system, or particle whose phase space is the cotangent bundle T ∗ R3 ∼ = R6 , 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 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 E over a ground field (typically R or C)) equipped with a bilinear and distributive multiplication ◦ . Then one defines a Jordan algebra (over R), as a a specific algebra over R for which: S◦T =T ◦S , , S ◦ (T ◦ S 2 ) = (S ◦ T ) ◦ S 2 , 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 defined as a real vector space UR together with a Jordan product ◦ and Poisson bracket { , }, satisfying : 1. for all S, T ∈ UR , S◦T =T ◦S {S, T } = −{T, S} 2. the Leibniz rule holds {S, T ◦ W } = {S, T } ◦ W + T ◦ {S, W } for all S, T, W ∈ UR , along with 3. the Jacobi identity : {S, {T, W }} = {{S, T }, W } + {T, {S, W }} ∗ hHamiltonianAlgebroidsi created: h2013-03-2i by: hbci1i version: h40815i Privacy setting: h1i hTopici h81P05i h81R15i h81R10i h81R05i h81R50i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 4. for some }2 ∈ R, there is the associator identity : (S ◦ T ) ◦ W − S ◦ (T ◦ W ) = 1 2 } {{S, W }, T } . 4 Thus, the canonical transformations of the Poisson sigma model phase space specified by the Jordan-Lie (Poisson) algebra (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 structure on the target space. 2