Hamiltonian algebroids

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.html

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

=

ℝ

3

of a classical, mechanical system, or particle whose phase space is the cotangent bundle

T

*

⁢

ℝ

3

≅

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

∘

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 (

http://planetmath.org/JordanBanachAndJordanLieAlgebras

) defined as a real vector space

U

ℝ

together with a

Jordan product

∘

and

Poisson bracket

{

,

}

, satisfying :

1.

for all

S

,

T

∈

U

ℝ

,

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

∈

U

ℝ

, along with

3.

the

Jacobi identity

:

{

S

,

{

T

,

W

}

}

=

{

{

S

,

T

}

,

W

}

+

{

T

,

{

S

,

W

}

}

4.

for some

ℏ

2

∈

ℝ

, there is the

associator

identity

:

(

S

∘

T

)

∘

W

-

S

∘

(

T

∘

W

)

=

1

4

⁢

ℏ

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

∘

, 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

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