Nilpotent integrability, reduction of dynamical systems and a third-order Calogero-Moser system

0.82 MB·English

by  A. Ibort

NILPOTENT INTEGRABILITY, REDUCTION OF
DYNAMICAL SYSTEMS AND A THIRD-ORDER
CALOGERO-MOSER SYSTEM
5 A. IBORT1,2, G. MARMO3, M. A. RODR´IGUEZ4, AND P. TEMPESTA2,4
1
0
2

n
a Abstract. We present an algebraic formulation of the notion of inte-
J grability of dynamical systems, based on a nilpotency property of its
3 flow: it can be explicitly described as a polynomial on its evolution pa-
2 rameter. Suchapropertyisestablishedinapurelygeometric–algebraic
language, in terms both of the algebra of all higher-order constants of

] the motion (named the nilpotent algebra of the dynamics), and of a
h
maximal Abelian algebra of symmetries (called a Cartan subalgebra of
p
the dynamics). It is shown that this notion of integrability amounts
-
h to the annihilator of the nilpotent algebra being contained in a Cartan
t subalgebra of the dynamics. Systems exhibiting this property will be
a
m said to be nilpotent integrable.
Ournotionofnilpotentintegrabilityoffersanewinsightintothein-
[
trinsic dynamical properties of a system, which is independent of any

1 auxiliary geometric structure defined on its phase space. At the same
v time,itextendsinanaturalwaytheclassicalconceptofcompleteinte-
5 grability for Hamiltonian systems.
7 An algebraic reduction procedure valid for nilpotent integrable sys-
8 tems, generalizing the well-known reduction procedures for symplectic
5
and/orPoissonsystemsonappropriatequotientspaces,isalsodiscussed.
0
Inparticular,itisshownthatalargeclassofnilpotentintegrablesys-
.
1 temscanbeobtainedbyreductionofhigher-orderfreesystems. Thecase
0 of the third-order free system is analyzed and a nontrivial set of third-
5 order Calogero-Moser-like nilpotent integrable equations is obtained.
1
:
v Contents
i
X
1. Introduction 2
r
a 2. Algebras, derivations and dynamical systems 6
2.1. Dynamical systems in smooth manifolds 6
2.2. Algebras of functions and differentiable algebras 7
2.3. Generating sets 9
2.4. Derivations and their flows on differentiable algebras 10
3. The algebraic formulation of nilpotent integrability 11
AI would like to thank the partial support by the Spanish MICIN grant MTM 2010-
21186-C02-02 and QUITEMAD+. GM wishes to thank the “Santander-UCIIIM Excel-
lence Chairs 2011” for supporting his stay at the University Carlos III de Madrid during
thedevelopmentofthisresearch. MARandPTwishtothankthesupportbytheresearch
project FIS2011–22566, Ministerio de Ciencia e Innovaci´on, Spain.
1
2 A.IBORT1,2,G.MARMO3,M.A.RODR´IGUEZ4,ANDP.TEMPESTA2,4
3.1. Infinitesimal symmetries 11
3.2. Constants of the motion 12
3.3. The nilpotent algebra of a derivation 13
3.4. The flow of a regular derivation on a differentiable algebra 15
3.5. Strict nilpotent integrability 17
3.6. Complete nilpotent integrability 18
4. Generalized algebraic reduction of nilpotent integrable systems 19
4.1. Restriction to invariant subspaces 19
4.2. Quotienting by invariant relations 20
4.3. Generalized reduction 20
4.4. Reduction of nilpotent integrable systems 21
5. Some explicit classes of nilpotent systems. The higher-order
Calogero-Moser systems 23
5.1. Linear systems 23
5.2. Reduction of higher-order free systems 23
6. Discussion and Future Perspectives 30
References 31
1. Introduction
In this paper, a new approach to the notion of integrability of dynamical
systems that extends the standard approach `a la Liouville, and also includes
such obvious “integrable” systems like uniformly accelerated ones, will be
discussed. For reasons that will become evident later on, this new notion of
integrability will be called nilpotent integrability.
One of the main features of nilpotent integrability is that it does not
require any auxiliary underlying geometrical structure for its formulation.
In this perspective, nilpotent integrability becomes an intrinsic property of
a dynamical system, and the specific geometry associated to it comes later.
Moreover, we shall show that nilpotent integrability is consistent with the
different procedures of reduction available for dynamical systems. They will
allow to create new nilpotent integrable systems out of simple examples.
The main motivation behind the idea that ‘integrability’, as a property of
a dynamical system, must be prior to any geometry associated to it, comes
fromamostpragmaticapproachtointegrability. Forinstance,ifwedemand
that a system is ‘integrable’ if its flow could be explicitly exhibited by using
a finite number of quadratures, there is no obvious geometrical content in
such a definition.
As a logical consequence of this observation, it is natural to try to refor-
mulate the well established notion of complete integrability of Hamiltonian
systemswithoutappealingtoanyauxiliarygeometry,suchasasymplecticor
a Poisson structure (see, for instance, [Mm85] and references therein). The
consistency of these ideas with general reduction procedures would reflect
NILPOTENT INTEGRABILITY AND REDUCTION 3
the fact that quadratures and elimination of variables should be compatible
with each other.
Toclarifythescopeoftheideasleadingtothenotionofnilpotentintegra-
bility, let us consider first the instance of a complete integrable Hamiltonian
systemsystemΓfromthestandardperspective. Thus,ifΓrepresentsacom-
plete integrable Hamiltonian system on a symplectic manifold, introducing
action-angle variables h , θj, j = 1,...,n, the vector field describing Γ will
j
look like
∂
Γ = νj(h) .
∂θj
The explicit expression of its flow ϕ , given by
t
h ϕ = h , θj ϕ = θj +νj(h)t,
j j t
◦ ◦
shows that the flow ϕ , described in action-angle variables, is affine in t.
t
Moreimportant,itistrivialtocheckthatthederivativesalongthedynamics
of the chosen coordinates are given by
(1) θj = νj(h); h = 0.
Γ Γ j
L L
The previous equations can also be interpreted by saying that the repre-
sentation in action-angle variables of the derivation Γ is nilpotent of order
2. It is relevant to point out that Eqns.(1) can be stated as a definition of
complete integrability without any reference to a symplectic structure.
Let us consider now simple examples of systems which are ‘integrable’
in an obvious sense but that do not fit naturally in the previous scheme.
...
For instance, let us consider a uniformly accelerated system in R3: r = 0,
r R.
∈
Theequationsofmotionofsuchasystemaredescribedbythevectorfield
on (r,v,a) T2R3
∈
∂ ∂
Γ = v +a ,
· ∂r · ∂v
with a a constant vector. Its flow ϕ has the explicit polynomial expression
t
1
r ϕ = r+vt+ at2.
t
◦ 2
Notice that the acceleration a is obviously a constant of the motion, while v
and r are higher-order constants of the motion, that is v = r˙, a = v˙ =¨r and
a˙ = 0. Similarly, the kinetic energy E = 1v2 is not anymore a constant of
1 2
the motion. However,
E = E˙ = v a, E = E˙ = a2, E˙ = 0,
2 1 3 2 3
·
i.e., E is a constant of the motion, while E , E are higher-order constants
3 2 1
of the motion (of orders 2 and 3 respectively). The same can be argued
with respect to the tower of (higher-order) constants of the motion derived
from the standard angular momentum: L = r v, L = L˙ = r a,
1 2 1
L = L˙ = v a, L˙ = 0. ∧ ∧
3 2 3
∧
4 A.IBORT1,2,G.MARMO3,M.A.RODR´IGUEZ4,ANDP.TEMPESTA2,4
The way how the previous ideas fit with the notion of reduction can be
nicely described by considering the basic example of the completely inte-
grable free system ¨r = 0 in R3. It is easy to construct other, perhaps more
interesting, completely integrable systems by reducing it using its integrals
of motion. For instance, by using the standard polar decomposition of r,
r = rn, n n = 1;
·
r˙ = r˙n+rn˙ , n n˙ = 0;
·
¨r = r¨n+2r˙n˙ +rn¨, n n¨ = n˙2,
· −
we may write its constants of the motion E and L as
1 1
E = 1 (cid:0)r˙2+r2n˙2(cid:1) , l2 = L2 = r4n˙2.
1 1 1
2
Then, once we restrict to the submanifold
Σ(k,α) = (r,v) TR3 αl2+2(1 α)E = k, 0 α 1 ,
1 1
{ ∈ | − ≤ ≤ }
since for our system we have r¨ = rn˙2, we get the following family of
completely (and explicitly) integrable reduced second-order systems in one-
dimension:
αl2+2(1 α)E (1 α)r˙2
r¨= 1 − 1− − .
r(αr2+(1 α))
− ...
It is immediate to generalize the previous construction to the system r = 0.
Now we have the additional relations
... ... ... ...
r = rn+3r¨n˙ +3r˙n¨ +rn , n n = 3n˙ n¨.
· − ·
... ...
Moreover, we have r = 3r˙n˙2 rn n. The towers of higher-order constants
− ·
of the motion E ,E ,E and L ,L ,L before defined, allow us to write e.g.
1 2 3 1 2 3
E = n˙ n¨ = r˙r¨+rr˙n˙2+r2n˙ n¨. Hence we get the explicitly integrable third
2
· ·
order system:
.r..= 3E2−r˙r¨, E (t) = a2t+c.
2
r
...
In other words, the reduction of systems like r = 0, possessing polynomial
flows, gives rise to systems that can be explicitly integrated and possess
polynomial flows as well. Additional examples of this kind will be discussed
later on.
Therefore, the main idea behind the notion of nilpotent integrability is to
takeadvantagenotonlyoftheconstantsofthemotion, butalsoofthewhole
familyofhigher-orderintegralsofmotionassociatedwiththesystem. Ifthis
family (that will be called the nilpotent algebra of the dynamical system)
is large enough, then the flow of the system will be shown to be explicitly
described,intermsoftheintegrals,asapolynomialfunctionoftime. Insuch
a case, the system will be said to be strictly nilpotent integrable. Systems
...
obtained by reduction of the system r = 0 will provide examples of the
strictly nilpotent case.
However, it could also happen that the nilpotent algebra associated with
a given system would not be large enough, although there exists an Abelian
NILPOTENT INTEGRABILITY AND REDUCTION 5
algebra of symmetries of the dynamics large enough to complement it. For
instance, in the case of completely integrable systems such Abelian algebra
of symmetries is the algebra of Hamiltonian vector fields defined by the ac-
tion variables, whose flow parameters are the “angle” variables θj. Such
systems will be called the completely nilpotent integrable ones. A partic-
ular instance of them are the standard completely integrable Hamiltonian
systems to which we have referred to.
Due to the paramount role played in this analysis by the algebraic struc-
ture of the families of higher-order constants of the motion and various Lie
algebras of symmetries of the system, we are led to develop the theory in
an algebraic framework, finding also inspiration in the notion of algebraic
integrability introduced in [La93].
It is important to mention that the algebraic framework also allows to
extend in a meaningful way the notion of ‘integrability” to quantum sys-
tems (see for instance [Cl09], [Fa10], and the discussion in [Ca14], Chaps.
7 and 8). In fact, the algebraic approach to the description of physical sys-
tems has gained a significant weight since the discovery of quantum group
symmetriesinphysicalsystems. Morerecentdevelopmentsinquantumgrav-
ity (see for instance [La97], [Va06], [Ba10]) point out the need for a non–
commutative description of space-time, hence emphasizing the algebraic ap-
proach to their study. In all these contexts, the algebraic view-point instead
of a set–theoretical description of physical systems is mandatory.
Hence, it is all but natural to pursue a foundation of the theory of dy-
namical systems from this point of view. Even more, in doing so, as the
following discussion will show, new insight and ideas can emerge naturally.
Consequently, we shall introduce the notions of strict and complete nilpo-
tent integrability of a dynamical system Γ by identifying it with a derivation
ofanalgebra(ofanappropriateclass) . Wewillprovethat,underadequate
F
conditions, the flow of such system can be represented as a polynomial func-
tion of time. As was mentioned above, our theory applies to a large class
of dynamical systems, possessing no obvious geometrical structure, like a
symplectic, or Poisson one, etc. The algebraic framework will also make
transparent the consistency of generalized reduction procedures (i.e. reduc-
tion with respect to subalgebras and ideals compatible with the dynamics)
with the notion of nilpotent integrability. Hence we shall prove that, under
appropriate hypotheses, if a system is nilpotent integrable its reductions are
nilpotent integrable too.
The formalism of differentiable spaces is the language we have chosen
to deal with the algebraic formulation of dynamics. Differentiable spaces
are one of the most useful extensions of the notion of differentiable mani-
folds that permit a simple algebraic description. In fact, local models for
differentiable spaces are just differentiable algebras, i.e., quotients of alge-
bras of smooth functions on open sets in finite dimensional Euclidean spaces
equippedwiththestrongWhitneytopology. Theuseofdifferentiablespaces
6 A.IBORT1,2,G.MARMO3,M.A.RODR´IGUEZ4,ANDP.TEMPESTA2,4
will also allow to keep the formalism close to the standard description of dy-
namical systems on manifolds. Another advantage of this approach is that
a large class of mild singularities are automatically dealt with within this
formalism. Also, some technical difficulties in dealing with local charts in
the standard formalism of differentiable manifolds are overlooked. It is also
useful to mention here that the present algebraic approach in principle can
be extended to the case of dynamical systems in the context of Galois dif-
ferential algebras along the lines discussed for instance in [Te13].
The structure of the paper is the following. The first subsections of Sec-
tion 2 will be devoted to review the standard notions of dynamical systems
both in the language of smooth manifolds and of differentiable spaces, mak-
ing the transition as smooth as possible. In Section 3, we shall discuss the
fundamental structures attached to a dynamical system, i.e., its algebra of
symmetriesanditsnilpotentalgebra, andwewillintroducethefundamental
notion of nilpotent integrability. In Section 4, a theory of generalized reduc-
tions will be formulated in the framework of nilpotent integrability. Finally,
in Section 5 a family of examples leading to a Calogero-Moser higher-order
nilpotent integrable system will be discussed.
2. Algebras, derivations and dynamical systems
2.1. Dynamical systems in smooth manifolds. Let M be a smooth
paracompact differentiable manifold of dimension n. Local coordinates on
M will be denoted by xi, i = 1,...,n. Let (M) be the commutative
F
algebra of smooth functions on M. Hereafter we will restrict to real valued
functions (although in some applications complex valued functions may also
be relevant).
DynamicsonM willbedescribedbyavectorfieldΓ,whichisaderivation
of the algebra (M), i.e. a linear map on (M) such that Γ(fg) = Γ(f)g+
F F
fΓ(g). The space of derivations of the algebra ( ), denoted in what
F M
follows by Der( ( )), is a module over the algebra ( ). Moreover, it
F M F M
converts into a Lie algebra if equipped with the Lie bracket
[Y ,Y ](f) = Y (Y (f)) Y (Y (f)), Y ,Y Der( ( )), f ( ).
1 2 1 2 2 1 1 2
− ∀ ∈ F M ∈ F M
The dynamics represented by Γ will be described in two possible ways.
a) As a system of autonomous first order differential equations, that in local
coordinates xi takes the form:
dxi
(2) = Γi(x), i = 1,...,m,
dt
where the vector field locally reads Γ = Γi(x)∂/∂xi.
b) Equivalently, as an evolution equation on (M):
F
df
(3) = Γ(f), f (M).
dt ∈ F
NILPOTENT INTEGRABILITY AND REDUCTION 7
The local flow of Γ will be denoted by ϕ and satisfies:
t
dϕ
t
(4) = Γ ϕ .
t
dt ◦
If the vector field is complete, the flow ϕ defines a one–parameter group
t
of automorphisms of the algebra (M) (a one–parameter family of auto-
F
morphisms for non–autonomous systems). Alternatively, we can integrate
formally eq. (3) to obtain:
(cid:88)∞ tn
(5) ϕ∗f = f ϕ = exp(tΓ)f = Γn(f).
t t
◦ n!
n=0
2.2. Algebras of functions and differentiable algebras. The algebra
(M)ofsmoothfunctionsonthemanifoldM isequippedwiththetopology
F
of uniform convergence on compact sets of the function and their deriva-
tives to order r (r = 1,2,..., ). However, for the purpose of extending
∞
the standard notions of differential calculus, it is necessary to consider the
strong or Whitney topology on (M), that is the topology defined by the
F
countable family of seminorms p (f) = ∂|k|f/∂xk , where K is
ik || ||L∞(Ki) { i}
a subcovering by compact sets of a small enough open covering U of M.
i
{ }
Thus (M) becomes a Fr´echet topological algebra, i.e., a locally multiplica-
F
tive convex algebra which is metrizable and complete, where multiplicative
convex means that the topology is defined by a countable family of submul-
tiplicative seminorms.
The algebra C∞(Rn) is an instance of a Fr´echet topological algebra and
the model for the notion of a differentiable algebra [Ma66] (see also [Na03]
for a recent account of the theory of differentiable algebras).
Definition 1. A differentiable algebra is a real Fr´echet algebra isomorphic
to C∞(Rm)/ , where is a closed ideal of C∞(Rm).
J J
Notice that the class of differentiable algebras is larger than the class of
algebras of differentiable functions on a manifold (for instance C∞(R)/(x2)
is not an algebra of differentiable functions on a smooth manifold).
BecauseofWhitney’sembeddingtheorem,anysmoothparacompactman-
ifold M can be embedded into Rm, hence (M) = C∞(Rm)/ , where
∼
is the closed ideal of smooth function on RFm vanishing on M J(as a closeJd
submanifold of it). Hence = C∞(M) is a differentiable algebra too.
F
Definition 2. Given a real topological algebra , we define its spectrum
SpecR( ) as the space of continuous R–morphiFsms ϕ: (cid:47)(cid:47) R equipped
F F
with the natural topology.
This space can be identified with the space of real maximal ideals of .
F
When M is a Hausdorff smooth manifold satisfying the second countabil-
ity axiom, then SpecR( (M)) ∼= M. Given an element f and a point
F ∈ F
x SpecR we define the Gel’fand transform fˆ(x) = x(f). Thus, any f
∈ F
8 A.IBORT1,2,G.MARMO3,M.A.RODR´IGUEZ4,ANDP.TEMPESTA2,4
defines a function on SpecR . If we consider (M), then fˆcoincides with
F F
f.
The spectrum of a differentiable algebra is the local model for a class of
spaces called ∞–spaces or differentiable spaces for short, that extend in a
C
natural way the notion of smooth manifolds (see for instance [Na03] and
references therein for a modern account of the foundations of the theory of
differentiable spaces).
Differentiable spaces are ringed spaces over a sheaf of differentiable alge-
bras. Theyallowtoworkwithalargeclassoftopologicalspaces,whichshare
with smooth manifolds most of their theoretical formulation and the differ-
ential calculus attached to them. At the same time, differentiable spaces
can exhibit singularities and other structures typically arising in the stan-
dard processes of reduction of dynamical systems. As was commented in
the introduction, this is the reason why we have chosen such a structure
to introduce the notion of nilpotent integrability. Actually, we will use in
this paper the “local” formulation of the theory only, i.e., the notion of dif-
ferentiable algebras. However, once formulated in the case of differentiable
algebras, nilpotent integrability can be extended naturally to the category
of differentiable spaces.
Differentiable spaces defined as the real spectrum of a differentiable al-
gebra are called affine differentiable spaces. It can be proved that a dif-
ferentiable space is affine if and only if it is separated. This means that at
eachpointthereisafinitelocalfamilyoffunctionsseparatinginfinitelynear
points (see later, Def. 2), with bounded dimension, and its topology has a
countable basis (see the “Embedding Theorem” at [Na03], p. 67). Thus,
any compact separated differentiable space is affine and a standard smooth
manifold can be characterized as a separated, finite-dimensional differen-
tiable space with a countable basis for its topology.
Given a differentiable algebra and a point x SpecR , let mx be the
F ∈ F
maximal ideal of all f such that fˆ(x) = 0. Also, let := be
∈ F Fx Fmx
the localization of with respect to the multiplicative system S defined
x
F
by m , namely the set of all elements f / m .
x x
∈
Definition 3. We shall call the elements on the germs of elements f of
x
F
at x.
F
They can be considered as residue classes of elements on with respect
F
to the ideal of elements of vanishing on open neighborhoods of x; we shall
F
denote them by [f] .
x
For a differentiable algebra , the space of its derivations Der( ) can
F F
be equipped with its canonical Lie algebra structure [ , ]. Such a space is
· ·
a –module. The space Der( )∗ of –valued –linear maps on Der( ) is
F F F F F
the space of 1–forms of the differentiable algebra . The space Der( )∗ is
F F
again a –module; we will denote it by Ω1( ) (or simply by Ω1 for short).
F F
Let U be an open set in a differentiable space X. All objects considered
so far can be localized to U in a standard way. Therefore, we can consider
NILPOTENT INTEGRABILITY AND REDUCTION 9
the differentiable algebras , Der( ) , Ω1, etc. Moreover, differentiable
FU F U U
spaces admit partitions of unity (see [Na03] p. 52).
The differential map d: (cid:47)(cid:47) Ω1 is defined as df(Y) = Y(f) for any
F
Y Der( ). In the case of = C∞(M), the space Ω1 is just the module
∈ F F
of smooth 1–forms Ω1(M) on the manifold M.
Alternatively, we can associate with any point x SpecR the linear
∈ F
spacemx/m2x denotedbyTx∗X whereX = SpecR . Again, thespaceTx∗X is
F
the localization of the module Ω1 with respect to the multiplicative system
defined by m . Then given f m , we have that df(x) := [df] can be
x x x
∈
identified with the class of f in m /m2. In the same vein it is possible to
x x
show that the dual space to T∗X, namely the tangent space T X to the
x x
differentiable space X at x, is just the space of derivations of the localized
algebra .
x
F
2.3. Generating sets.
Definition 4. Given a differentiable algebra , we shall say that a family
F
of functions g1,...,gn separate points infinitely near to x X = SpecR if
∈ F
the differentials dg (x),...,dg (x) generate T∗X as a linear space.
1 n x
Equivalently, the differentials dg (x) of the family separate elements v in
k x
T X, i.e., for any v = w there is g such that v (g ) = w (g ).
x x x k x k x k
(cid:54) (cid:54)
Definition 5. We say that the family of functions = g of the differ-
α
G { }
entiable algebra is a differential generating set if they separate points in
F
X = SpecRF and if for any x ∈ X, there is a finite subfamily {gα1,...,gαn}
of that separate infinitely near points to x.
G
Noticethatif isadifferentialgeneratingsetforthedifferentiablealgebra
G
,thenforeachpairofderivationsY andY thereexistsatleastoneelement
1 2
F
g such that Y (g ) = Y (g ).
α 1 α 2 α
∈ G (cid:54)
When the previous condition holds, we say that the functions g sepa-
α
rate derivations. It is not hard to see that a large class of manifolds, for
instance C∞ paracompact second countable manifolds, possesses generating
sets. Also,if isageneratingsetforthealgebra asanassociativealgebra,
G F
then it is a differential generating set for .
F
Since derivations can be localized, generating sets also separate local
derivations. This fact is used to prove the following important property
of differential generating sets, which justifies their name.
Lemma 1. Let M be a smooth manifold. A set of functions = g of
α
G { }
(M) is a differential generating set iff the set of 1–forms d = dg gen-
α
F G { }
erates the algebraic dual of the space of derivations Der( )∗, which coincides
F
with the space Ω1(M) of smooth 1–forms on M.
Proof. Consider first an open set U contained in the domain of a local chart
of M. If there were a local 1–form σ that could not be written as σ =
U U
σ dg , then the span of dg would be a proper subspace of T∗U and it
U,α α α
would exista vector fieldX lying in the annihilator ofsuchsubspace. Hence
10 A.IBORT1,2,G.MARMO3,M.A.RODR´IGUEZ4,ANDP.TEMPESTA2,4
dg (X) = 0 for all α. This argument can be made global by using partitions
α
of the unity: for any 1–form σ on M it must exist a family of functions σ
α
with compact support on M such that σ = σ dg . (cid:3)
α α
In the subsequent analysis, we shall assume the existence of a differential
generating set for . Let us mention another direct consequence of the
G F
properties of a generating set . Since locally the differentials dg generate
α
G
T∗X with X the spectrum of , we can extract a subset dg that is locally
F αi
independent, i.e. such that dg dg = 0 on a open neighborhood of
α1∧···∧ αm (cid:54)
any given point. Therefore, we will say that the functions g define a local
α
k
coordinate system. Later on we will use this fact to write explicit formulas
in terms of subsets of generating sets.
We will introduce a notion of finiteness for algebras which is funda-
F
mental in order to define our concept of integrability.
Definition 6. The differentiable algebra is said to be of finite type if it
admits a finite differential generating set F= g ,...,g ,N N.
1 N
G { } ∈
This more restrictive condition is satisfied for instance if the manifold is
compact or of finite type. Indeed, in such case it can be embedded into a
finite dimensional Euclidean space whose coordinate functions, restricted to
the embedded manifold, provide a differential generating set. As a conse-
quence of the previous discussion, it can be shown that if M is a smooth
manifold, a differential generating set for the differentiable algebra (M)
F
provides a set of local coordinate systems, i.e., an atlas for the manifold
M, by restricting to small enough open sets and shieving out dependent
functions.
2.4. Derivations and their flows on differentiable algebras. Let Γ
be a derivation of the differentiable algebra of finite type , hence =
∼
∞(Rn)/ . Denoting by X, as before, the real spectrumFof , for Feach
C J F
x X the derivation Γ defines an element Γ T X. Hence, X is a closed
x x
diff∈erentiable subspace of Rn, and the canonica∈l injection i: X (cid:47)(cid:47)Rn maps
Γ to a tangent vector i Γ T Rn. Moreover, we can extend the vector
x ∗ x i(x)
∈
field i∗Γ along i(X) to a vector field Γ(cid:101) in Rn. Let ϕ˜t be the flow of Γ(cid:101). By
construction ϕ˜ leaves i(X) invariant. Let us denote by ϕ the restriction
t t
of ϕ˜ to X (that always exists because of the universal property of closed
t
differentiable subspaces [Na03] p. 60). We will denote by ϕ the flow of the
t
derivation Γ. The flow ϕ will act on elements f as
t
∈ F
ϕ∗(f) = ϕ˜∗(f˜)+ , f˜+ = f.
t t
J J
Also, the flow ϕ can be integrated formally by using a close analog to
t
formula (5):
(cid:88)∞ tn
(6) ϕ∗(f) = Γ(cid:101)n(f˜)+ .
t
n! J
n=0