Chapter 3 Symplectic reduction on locally Euclidean Frˆlicher spaces
In this chapter, we present an extension of the fundamentals of the « Marsden-Weinstein quotient » (also called reduced space) that is well known in the category of nite dimensional smooth manifolds. The process producing a reduced space is named the symplectic reduction (see [22; 77]). The modeling space for the extension of the symplectic reduction process is the nite dimensional locally Euclidean Frˆlicher Space. We say just space for Locally Euclidean Frˆlicher Space. For a space of the category under consideration we dene a free, proper and Hamiltonian action of a Frˆlicher Lie group provided with an equivariant moment map, as in the case of the category of smooth manifolds (see [8; 36; 66; 85; 112]). The result is a quotient that is still a nite dimensional space with a symplectic structure induced from the original one on the ambient space (see [8; 112]). And then, we raise some fundamental questions such as:
- -Why the properness and the freeness of the action?
- -Why is the compactness assumption for the operating Frˆlicher-Lie group so relevant?
- -What happens when one drops one of the previous conditions comparatively to the category of smooth manifolds?
Symplectic locally Euclidean Frˆlicher spaces
The symplectic framework in the category Frl was introduced in , the F-bundles in  and the symplectic reduction process on locally Euclidean Frˆlicher spaces was investigatedin . The main references on symplectic linear spaces and symplectic manifolds should be [1; 47; 70]. We introduce below the symplectic structure on linear locally Euclidean Frˆlicher spaces. Thereafter, we will present the general non linear case.
Denition 3.1.1. Let M be a nite dimensional linear F-space and ! a F-smooth 2-form on M. The form ! is called a symplectic form or a symplectic structure on M if it is both skew-symmetric and non-degenerate.
That is, !(x; y) + !(y; x) = 0 for all x; y 2 M and !(x; y) = 0 for all y 2 M implies that x = 0. The pair (M; !) is called a symplectic linear F-space.
For an exterior 2-form denoted by !, the induced linear map ![ : M ! M is dened by [(x) = x! = !(x; :), for any x 2 M. That is, for all y 2 M, The map ![ is an isomorphism if, and only if the 2-form ! is non-degenerate if, and only if rank(!) = dim M = n:
Remark 3.1.1. The non-degeneracy of the symplectic form ! is equivalent to the following statements.
The linear map ![ : M ! M , as dened in Equation (3:1), is a smooth isomorphism of linear F-spaces.
The transpose !t = ! is non-degenerate.
- The dimension dim M = dim ![(M) = rank(!) = 2p, that is maximal even integer, where p is independent of the choice of a basis in M.
- There is a basis fu1; : : : ; up; v1; : : : ; vpg in M, such that !(ui; uj) = !(vi; vj) = 0 and !(ui; vj) = ij, where i; j 2 f1; 2; : : : ; pg and ij is the Kronecker symbol. This basis is called the canonical or symplectic basis.
- Let (!ij)1<i;j<p be the matrix of ! in any basis and (!ijt)1<i;j<p its transpose. It follow that det(!ij)1<i;j<p 6= 0 and also det(!ijt)1<i;j<p 6= 0. Moreover rank(!ij)1<i;j<p = 2p.
It follows that any symplectic linear space is even dimensional.
Denition 3.1.2. Let (M; !) be a symplectic linear F-space of dimension n. Let W and W be two linear subspaces of M. Two vectors x and y in M are called orthogonal with regard to ! ( or !-orthogonal ) if !(x; y) = 0. The linear subspaces W and W 0 are called !-orthogonal if every x2W is !-orthogonal to every y 2W 0.
The set fx 2 M j !(x; y) = 0 for every y 2 Wg := orth!W := W? := N! -orthogonal of W and it is the maximal element in the set of all linear subspaces of M which are !-orthogonal to W .
Equation (3:1) and the rst item in Remark 3:1:1 allow the following denition.
Denition 3.1.3. Let (M; !) be a symplectic linear F-space of dimension 2n and N its linear
F-subspace of dimension s. Let !N and ![ be the restrictions of ! and ![ to N, respectively.
The kernel of !N is given by Ker!N = Ker![ = fx 2 N j ![(x) = x! = 0g = N \ N?. N
The kernel of !N is not necessarily equal to f0g. It raises the need of a characterization among linear F-subspaces of M with regard to !N as in the forthcoming Denition.
Denition 3.1.4. Let (M; !) be a symplectic linear F-space of dimension 2n and N its linear F-subspace of dimension s. Let orth!W as dened in Denition 3:1:2.
The linear F-subspace N is called symplectic if !N is a symplectic structure on N dened by !N := N !, where N is the canonical inclusion of N into M. That is, if N \ orth!N = f0g. The linear F-subspace N is called isotropic if !N = 0. That is, if N orth!N.
The linear F-subspace N is called co-isotropic if !orth!N = 0. That is, if orth!N N. The linear F-subspace N is called Lagrangian if N is both isotropic and co-isotropic. That is, N = orth!N.
Proposition 3.1.1. If M is a linear F-space of dimension m and ! any 2-form (skew sym-metric) on M with N = Ker!, then there exists a symplectic form ! on the linear F-quotient space M=N, such that the form ! is the pullback ! of !, where is the canonical projection of M onto the quotient.
Proposition 3.1.2. Let (M; !) be a symplectic linear F-space of dimension 2n and N its linear F-subspace. The formula !N := N ! = N !N denes a symplectic form, on the quotient linear F-space N = N=(N \ orth!N), induced by !, where is the canonical projection of M onto the quotient and N the canonical inclusion of N into M.
Denition 3.1.5. The symplectic linear F-space (N ; !N ) is called the reduced symplectic linear F-space associated to N.
1 Introduction 1
1.1 Review of literature
1.2 Thesis outline
2 Frölicher spaces
2.1 Basic concepts on F-spaces
2.2 Locally Euclidean Frölicher spaces
2.3 Locally Euclidean F-subspace
2.4 Tangent spaces on F-spaces
2.5 Exterior algebra on F-spaces
3 Symplectic reduction on locally Euclidean Frölicher spaces
3.1 Symplectic locally Euclidean Frölicher spaces
3.2 Flows, Integral curve and Exponential map
3.4 Moment map
3.5 Symplectic reduction on locally Euclidean Frölicher spaces
4 Topological inheritance on symplectic quotients of ringed Frölicher spaces
4.1 Ringed F-space
4.2 Smooth Gelfand representation
4.4 Hausdor and paracompactness inheritance
5 Sheaf and cohomology on reduced spaces of locally Euclidean F-spaces
5.1 Sheaves, Presheaves and Properties
5.2 Alexander-Spanier cohomology
5.3 de Rham cohomology
5.4 Singular cohomology
5.5 Cech cohomology
5.6 Multiplicative structure and de Rham theorem
5.7 Isomorphism on reduced space
6 Modern Formalism of mechanics
6.1 Poisson geometry
6.2 Hamiltonian systems
6.3 Equivariance of moment map
6.4 Symplectic reduction of dynamical systems
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COHOMOLOGIES ON SYMPLECTIC QUOTIENTS OF LOCALLY EUCLIDEAN FRÖLICHER SPACES