Talk in Potsdam, Germany

1. Preamble
Localization
Deformation Quantization
Results
Localization and Lie Rinehart algebras in deformation
quantization
Hamilton ARAUJO
Preprint on ArXiv: 2010.15701
Join work with Martin BORDEMANN (Phd supervisor) and
Benedikt HURLE
Université de Haute-Alsace
IRIMAS - Departément de Mathematiques
Arbeitsgruppenseminar Analysis
Universität Potsdam, Deutschland, 04 dez 2020
1/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

2. Preamble
Localization
Deformation Quantization
Results
Preamble
Noncommutative localization
for Star products
Deformation quantization Localization
2/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

3. Preamble
Localization
Deformation Quantization
Results
Preamble
Noncommutative localization
for Star products
Deformation quantization Localization
In this work we talk about:
2/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

4. Preamble
Localization
Deformation Quantization
Results
Preamble
Noncommutative localization
for Star products
Deformation quantization Localization
In this work we talk about:
Algebraic localization
Analytic localization
2/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

5. Preamble
Localization
Deformation Quantization
Results
Preamble
Noncommutative localization
for Star products
Deformation quantization Localization
In this work we talk about:
Algebraic localization
Analytic localization
Compare this two types of localization.
2/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

6. Preamble
Localization
Deformation Quantization
Results
Preamble
Noncommutative localization
for Star products
Deformation quantization Localization
In this work we talk about:
Algebraic localization
Analytic localization
Compare this two types of localization.
⇒ Noncommutative localization is not
very well know.
2/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

7. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

8. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

9. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

10. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

11. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
Deformation quantization
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

12. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
Deformation quantization
- Star products
- Concrete localization
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

13. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
Deformation quantization
- Star products
- Concrete localization
Results
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

14. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
Deformation quantization
- Star products
- Concrete localization
Results
- and comments
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

15. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

16. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

17. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
Consider the following equivalence relation on R × (R {0R}):
(r, s) ∼ (r′
, s′
) ⇐⇒ rs′
= r′
s.
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

18. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
Consider the following equivalence relation on R × (R {0R}):
(r, s) ∼ (r′
, s′
) ⇐⇒ rs′
= r′
s.
The quotient R =
R × (R {0R})
∼
= {(r, s) = r
s , r ∈ R and s ∈ S} -with the
usual operations of sum and product of classes- is called field of fractions of R
(e.g. R = Q or R = R(x)).
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

19. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
Consider the following equivalence relation on R × (R {0R}):
(r, s) ∼ (r′
, s′
) ⇐⇒ rs′
= r′
s.
The quotient R =
R × (R {0R})
∼
= {(r, s) = r
s , r ∈ R and s ∈ S} -with the
usual operations of sum and product of classes- is called field of fractions of R
(e.g. R = Q or R = R(x)).
Note: R is naturally included in R
x 7→
x
1
and the elements of R {0R} have become
invertible in R,
x
1
−1
=
1
x
.
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

20. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
Consider the following equivalence relation on R × (R {0R}):
(r, s) ∼ (r′
, s′
) ⇐⇒ rs′
= r′
s.
The quotient R =
R × (R {0R})
∼
= {(r, s) = r
s , r ∈ R and s ∈ S} -with the
usual operations of sum and product of classes- is called field of fractions of R
(e.g. R = Q or R = R(x)).
Note: R is naturally included in R
x 7→
x
1
and the elements of R {0R} have become
invertible in R,
x
1
−1
=
1
x
. The morphism x 7→
x
1
is in general not injective.
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

21. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

22. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
K is a commutative associative unital ring such that 1 = 1K ̸= 0 = 0K.
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

23. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
K is a commutative associative unital ring such that 1 = 1K ̸= 0 = 0K.
We will consider associative and unital K-algebras. We shall include unital
K-algebras isomorphic to {0} (for which 1 = 0).
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

24. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
K is a commutative associative unital ring such that 1 = 1K ̸= 0 = 0K.
We will consider associative and unital K-algebras. We shall include unital
K-algebras isomorphic to {0} (for which 1 = 0).
Def.: If R is a K-algebra, S ⊂ R is called multiplicative subset if 1 ∈ S and for all
s, s′ ∈ S we have ss′ ∈ S.
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

25. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
K is a commutative associative unital ring such that 1 = 1K ̸= 0 = 0K.
We will consider associative and unital K-algebras. We shall include unital
K-algebras isomorphic to {0} (for which 1 = 0).
Def.: If R is a K-algebra, S ⊂ R is called multiplicative subset if 1 ∈ S and for all
s, s′ ∈ S we have ss′ ∈ S.
Def.: A K-algebra morphism ϕ : R → R′ is called S-inverting if ϕ(S) ⊂ U(R′), where
U(R′) denote the group of invertible elements of R′.
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

26. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

27. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
Let R be a commutative K-algebra and S ⊂ R a multiplicative subset, then the
following binary relation ∼ on R × S defined by
(r1, s1) ∼ (r2, s2) if and only if ∃ s ∈ S : r1s2s = r2s1s
is an equivalence relation.
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

28. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
Let R be a commutative K-algebra and S ⊂ R a multiplicative subset, then the
following binary relation ∼ on R × S defined by
(r1, s1) ∼ (r2, s2) if and only if ∃ s ∈ S : r1s2s = r2s1s
is an equivalence relation.
RS :=
R × S
∼
is a commutative K-algebra, also called the quotient algebra or
algebra of fractions of R with respect to S.
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

29. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
Let R be a commutative K-algebra and S ⊂ R a multiplicative subset, then the
following binary relation ∼ on R × S defined by
(r1, s1) ∼ (r2, s2) if and only if ∃ s ∈ S : r1s2s = r2s1s
is an equivalence relation.
RS :=
R × S
∼
is a commutative K-algebra, also called the quotient algebra or
algebra of fractions of R with respect to S.
The equivalence classes (r, s) = r
s are also called fractions.
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

30. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
Let R be a commutative K-algebra and S ⊂ R a multiplicative subset, then the
following binary relation ∼ on R × S defined by
(r1, s1) ∼ (r2, s2) if and only if ∃ s ∈ S : r1s2s = r2s1s
is an equivalence relation.
RS :=
R × S
∼
is a commutative K-algebra, also called the quotient algebra or
algebra of fractions of R with respect to S.
The equivalence classes (r, s) = r
s are also called fractions.
There is a ring homomorphism (the numerator morphism) η(R,S) = η : R → RS
given by r 7→ r
1. This map defines a K-algebra stucture of RS.
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

31. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
The construction of the K-algebra RS gives us important properties.
7/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

32. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
The construction of the K-algebra RS gives us important properties.
Proposition:
If R is a commutative K-algebra and
S ⊂ R is a multiplicative subset we have:
a. η(R,S)(S) ⊂ U(RS).
b. Every element of RS is written as a
fraction η(r)η(s)−1, for some r ∈ R
and s ∈ S.
c. ker(η(R,S)) = {r ∈ R | rs = 0
for some s ∈ S}.
7/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

33. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
The construction of the K-algebra RS gives us important properties.
Proposition:
If R is a commutative K-algebra and
S ⊂ R is a multiplicative subset we have:
a. η(R,S)(S) ⊂ U(RS).
b. Every element of RS is written as a
fraction η(r)η(s)−1, for some r ∈ R
and s ∈ S.
c. ker(η(R,S)) = {r ∈ R | rs = 0
for some s ∈ S}.
Moreover, for S ⊂ R as before:
7/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

34. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
The construction of the K-algebra RS gives us important properties.
Proposition:
If R is a commutative K-algebra and
S ⊂ R is a multiplicative subset we have:
a. η(R,S)(S) ⊂ U(RS).
b. Every element of RS is written as a
fraction η(r)η(s)−1, for some r ∈ R
and s ∈ S.
c. ker(η(R,S)) = {r ∈ R | rs = 0
for some s ∈ S}.
Moreover, for S ⊂ R as before:
The pair (RS, η(R,S)) is universal
in the sense that for any S-inverting
morphism of commutative unital K-algebras
α : R → R′ uniquely factorizes, i.e.
R
η
//
α
RS
f
R′
where f is a morphism of unital K-algebras
determined by α (Universal property).
7/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

35. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

36. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
KAlgMS KAlg
(R, S)
S ⊂ R mult. subset R
ϕ : (R, S) → (R′, S′)
ϕ(S) ⊂ S′ ϕ : R → R′
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

37. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
KAlgMS KAlg
(R, S)
S ⊂ R mult. subset R
ϕ : (R, S) → (R′, S′)
ϕ(S) ⊂ S′ ϕ : R → R′
There is an obvious functor
U : KAlg → KAlgMS given by
U(R) = (R, U(R)) and, for the
commutative case, we already get a
localization functor L(R, S) = RS.
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

38. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
KAlgMS KAlg
(R, S)
S ⊂ R mult. subset R
ϕ : (R, S) → (R′, S′)
ϕ(S) ⊂ S′ ϕ : R → R′
There is an obvious functor
U : KAlg → KAlgMS given by
U(R) = (R, U(R)) and, for the
commutative case, we already get a
localization functor L(R, S) = RS.
(R, S)
| {z }
L
−
−
−
−
−
−
−
→ RS
|{z}
KAlgMS KAlg
z }| {
(R, U(R))
←
−
−
−
−
−
−
U
z}|{
R
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

39. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
KAlgMS KAlg
(R, S)
S ⊂ R mult. subset R
ϕ : (R, S) → (R′, S′)
ϕ(S) ⊂ S′ ϕ : R → R′
There is an obvious functor
U : KAlg → KAlgMS given by
U(R) = (R, U(R)) and, for the
commutative case, we already get a
localization functor L(R, S) = RS.
(R, S)
| {z }
L
−
−
−
−
−
−
−
→ RS
|{z}
KAlgMS KAlg
z }| {
(R, U(R))
←
−
−
−
−
−
−
U
z}|{
R
Proposition:
The functor L also exists in the
noncommutative case and L is left adjoint to U.
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

40. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

41. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

42. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
=⇒ The kernel ker(η) can vary: η as well as RS may be trivial although 0 ̸∈ S.
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

43. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
=⇒ The kernel ker(η) can vary: η as well as RS may be trivial although 0 ̸∈ S.
To get rid of these problems: assume that each left fraction η(s)
−1
η(r)
becomes a right fraction η(r′) η(s′)
−1
implying the condition:
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

44. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
=⇒ The kernel ker(η) can vary: η as well as RS may be trivial although 0 ̸∈ S.
To get rid of these problems: assume that each left fraction η(s)
−1
η(r)
becomes a right fraction η(r′) η(s′)
−1
implying the condition:
∀ (r, s) ∈ R × S ∃ (r′
, s′
) ∈ R × S : η(rs′
) = η(sr′
).
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

45. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
=⇒ The kernel ker(η) can vary: η as well as RS may be trivial although 0 ̸∈ S.
To get rid of these problems: assume that each left fraction η(s)
−1
η(r)
becomes a right fraction η(r′) η(s′)
−1
implying the condition:
∀ (r, s) ∈ R × S ∃ (r′
, s′
) ∈ R × S : η(rs′
) = η(sr′
).
This will motivate the following:
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

46. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Ore conditions [Øystein Ore] (1931)
Let R be a unital K-algebra and S ⊂ R be a multiplicative subset.
10/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

47. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Ore conditions [Øystein Ore] (1931)
Let R be a unital K-algebra and S ⊂ R be a multiplicative subset.
Def.: S is called a right denominator set if
a. For all r ∈ R and s ∈ S there are
r′ ∈ R and s′ ∈ S such that
rs′ = sr′
(S right permutable or right Ore
set),
b. For all r ∈ R and for all s′ ∈ S: if
s′r = 0 then there is s ∈ S such that
rs = 0 (S right reversible).
10/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

48. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Ore conditions [Øystein Ore] (1931)
Let R be a unital K-algebra and S ⊂ R be a multiplicative subset.
Def.: S is called a right denominator set if
a. For all r ∈ R and s ∈ S there are
r′ ∈ R and s′ ∈ S such that
rs′ = sr′
(S right permutable or right Ore
set),
b. For all r ∈ R and for all s′ ∈ S: if
s′r = 0 then there is s ∈ S such that
rs = 0 (S right reversible).
Def.: ŘS with η̌(R,S) = η̌ : R → ŘS
is said to be a right K-algebra of fractions of
(R, S) if:
a. η̌(R,S) is S-inverting,
b. Every element of ŘS is of the form
η̌(r) η̌(s)
−1
for r ∈ R and s ∈ S;
c. ker(η̌) = {r ∈ R | rs = 0, for some s ∈ S}
=: I(R,S) =: I.
10/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

49. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

50. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

51. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
2 If this is the case each such pair (ŘS, η̌) is universal and then each ŘS is
isomorphic to the canonical localized algebra RS.
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

52. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
2 If this is the case each such pair (ŘS, η̌) is universal and then each ŘS is
isomorphic to the canonical localized algebra RS.
3 Each ŘS is isomorphic to the quotient set RS−1 := (R × S)/ ∼ with respect to
the following generalized equivalence relation ∼ on R × S
(r1, s1) ∼ (r2, s2) ⇔ ∃b1, b2 ∈ R s.t. s1b1 = s2b2 ∈ S and r1b1 = r2b2 ∈ R.
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

53. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
2 If this is the case each such pair (ŘS, η̌) is universal and then each ŘS is
isomorphic to the canonical localized algebra RS.
3 Each ŘS is isomorphic to the quotient set RS−1 := (R × S)/ ∼ with respect to
the following generalized equivalence relation ∼ on R × S
(r1, s1) ∼ (r2, s2) ⇔ ∃b1, b2 ∈ R s.t. s1b1 = s2b2 ∈ S and r1b1 = r2b2 ∈ R.
Note: The proof of this theorem is quite complicated. We can find in [D.S.
Passman] (1980) a more direct proof.
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

54. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
2 If this is the case each such pair (ŘS, η̌) is universal and then each ŘS is
isomorphic to the canonical localized algebra RS.
3 Each ŘS is isomorphic to the quotient set RS−1 := (R × S)/ ∼ with respect to
the following generalized equivalence relation ∼ on R × S
(r1, s1) ∼ (r2, s2) ⇔ ∃b1, b2 ∈ R s.t. s1b1 = s2b2 ∈ S and r1b1 = r2b2 ∈ R.
Note: The proof of this theorem is quite complicated. We can find in [D.S.
Passman] (1980) a more direct proof. Moreover, RS−1 carries a canonical unital K-
algebra structure. In terms of the equivalences classes r1s−1
1 and r2s−1
2 we have:
r1s−1
1 + r2s−1
2 = (r1c1 + r2c2)s−1 and (r1s−1
1 )(r2s−1
2 ) = (r1r′)(s2s′)−1 where
s1c1 = s2c2 = s ∈ S (c1 ∈ S and c2 ∈ R) and r2s′ = s1r′ (s′ ∈ S and r′ ∈ R).
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

55. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Deformation Quantization
12/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

56. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
For a K-vector space V let V [[λ]] = {v =
P∞
i=0 viλi with vi ∈ V } be the
K[[λ]]-module of formal power series. Here K ∈ {R, C}.
13/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

57. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
For a K-vector space V let V [[λ]] = {v =
P∞
i=0 viλi with vi ∈ V } be the
K[[λ]]-module of formal power series. Here K ∈ {R, C}.
For a smooth diff. manifold X we write C∞(X) = C∞(X, K).
13/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

58. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
For a K-vector space V let V [[λ]] = {v =
P∞
i=0 viλi with vi ∈ V } be the
K[[λ]]-module of formal power series. Here K ∈ {R, C}.
For a smooth diff. manifold X we write C∞(X) = C∞(X, K).
Def.: A (formal) star product ∗
on a manifold X is a K[[λ]]-bilinear associative
operation C∞(X)[[λ]] × C∞(X)[[λ]] → C∞(X)[[λ]]
satisfying the following properties for all
f, g ∈ C∞(X):
1 ∗ f = f ∗ 1 = f,
f ∗ g = f · g + O(λ),
f ∗ g =
P∞
k=0 Ck(f, g)λk,
where Ck : C∞(X) ⊗ C∞(X) → C∞(X) are
bidifferential operators.
13/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

59. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
For a K-vector space V let V [[λ]] = {v =
P∞
i=0 viλi with vi ∈ V } be the
K[[λ]]-module of formal power series. Here K ∈ {R, C}.
For a smooth diff. manifold X we write C∞(X) = C∞(X, K).
Def.: A (formal) star product ∗
on a manifold X is a K[[λ]]-bilinear associative
operation C∞(X)[[λ]] × C∞(X)[[λ]] → C∞(X)[[λ]]
satisfying the following properties for all
f, g ∈ C∞(X):
1 ∗ f = f ∗ 1 = f,
f ∗ g = f · g + O(λ),
f ∗ g =
P∞
k=0 Ck(f, g)λk,
where Ck : C∞(X) ⊗ C∞(X) → C∞(X) are
bidifferential operators.
Example in C∞(R2)[[λ]]
In coordinates (x, p) the following
formula defines a star product for
f, g ∈ C∞(R2):
f ∗ g =
∞
X
k=0
λk
k!
∂kf
∂pk
∂kg
∂xk
(Multiplication of diff. operators)
13/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

60. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Deformation Quantization was founded by the seminal article [Bayen, Flato,
Frønsdal, Lichnerowicz, Sternheimer] (1978) and has become a large research area
covering several algebraic theories.
14/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

61. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Deformation Quantization was founded by the seminal article [Bayen, Flato,
Frønsdal, Lichnerowicz, Sternheimer] (1978) and has become a large research area
covering several algebraic theories.
The article by [Kontsevich] (1997) shows that important constructions are
possible, especially for Poisson manifolds. (C1(f, g) − C1(g, f) = {f, g}).
14/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

62. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Deformation Quantization was founded by the seminal article [Bayen, Flato,
Frønsdal, Lichnerowicz, Sternheimer] (1978) and has become a large research area
covering several algebraic theories.
The article by [Kontsevich] (1997) shows that important constructions are
possible, especially for Poisson manifolds. (C1(f, g) − C1(g, f) = {f, g}).
However, this theory does not play an important role in this job.
14/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

63. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Localization on open sets
15/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

64. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

65. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
Let ∗ =
P∞
k=0 λkCk be a star-product on a manifold X.
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

66. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
Let ∗ =
P∞
k=0 λkCk be a star-product on a manifold X.
We set K = K[[λ]] and R = C∞(X)[[λ]], ∗
.
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

67. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
Let ∗ =
P∞
k=0 λkCk be a star-product on a manifold X.
We set K = K[[λ]] and R = C∞(X)[[λ]], ∗
.
Let be Ω ⊂ X a open set and RΩ := C∞(Ω, K)[[λ]].
Let ∗Ω =
P∞
k=0 λkCk|Ω (the restriction of bidifferential operators to open sets is
well-defined!) =⇒ ∗Ω is a star-product on RΩ.
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

68. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
Let ∗ =
P∞
k=0 λkCk be a star-product on a manifold X.
We set K = K[[λ]] and R = C∞(X)[[λ]], ∗
.
Let be Ω ⊂ X a open set and RΩ := C∞(Ω, K)[[λ]].
Let ∗Ω =
P∞
k=0 λkCk|Ω (the restriction of bidifferential operators to open sets is
well-defined!) =⇒ ∗Ω is a star-product on RΩ.
It is clear that there is a morphism between unital K-algebras:
ηΩ = η :
R → RΩ
f 7→ f|Ω
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

69. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

70. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

71. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

72. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
Consider then the noncommutative abstract
localization
RS
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

73. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
Consider then the noncommutative abstract
localization
RS
Concrete localization
The space of all formal power series
only defined in Ω already provides
us with the K-algebra
RΩ = (C∞
(Ω)[[λ]], ⋆Ω)
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

74. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
Consider then the noncommutative abstract
localization
RS
Concrete localization
The space of all formal power series
only defined in Ω already provides
us with the K-algebra
RΩ = (C∞
(Ω)[[λ]], ⋆Ω)
Question:
RS
?
∼
= RΩ
Are these algebras isomorphic??
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

75. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
Consider then the noncommutative abstract
localization
RS
Concrete localization
The space of all formal power series
only defined in Ω already provides
us with the K-algebra
RΩ = (C∞
(Ω)[[λ]], ⋆Ω)
Question:
RS
?
∼
= RΩ
Are these algebras isomorphic??
Of course, look at next page.
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

76. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Results
18/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

77. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

78. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

79. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

80. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
2 As an immediate consequence we have that S is a right denominator set.
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

81. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
2 As an immediate consequence we have that S is a right denominator set.
3 This implies in particular that the algebraic localization RS
∼
= RS−1 of R with
respect to S is isomorphic to the concrete localization RΩ as unital
K-algebras.
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

82. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
2 As an immediate consequence we have that S is a right denominator set.
3 This implies in particular that the algebraic localization RS
∼
= RS−1 of R with
respect to S is isomorphic to the concrete localization RΩ as unital
K-algebras.
We don’t directly prove that S is right denominator set. This will follow from the
general theorem of localization.
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

83. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
2 As an immediate consequence we have that S is a right denominator set.
3 This implies in particular that the algebraic localization RS
∼
= RS−1 of R with
respect to S is isomorphic to the concrete localization RΩ as unital
K-algebras.
We don’t directly prove that S is right denominator set. This will follow from the
general theorem of localization.
The idea of the proof is to show the three conditions for (RΩ, ∗Ω, η) to be a right
K-algebra of fractions.
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

84. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

85. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
1 For g ∈ S ⊂ R and γ = η(g) we try to construct ψ ∈ RΩ; γ ∗Ω ψ = 1.
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

86. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
1 For g ∈ S ⊂ R and γ = η(g) we try to construct ψ ∈ RΩ; γ ∗Ω ψ = 1.
2 We work order by order. For k = 0 it’s easy (x 7→ ψ0(x) = γ0(x)−1
).
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

87. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
1 For g ∈ S ⊂ R and γ = η(g) we try to construct ψ ∈ RΩ; γ ∗Ω ψ = 1.
2 We work order by order. For k = 0 it’s easy (x 7→ ψ0(x) = γ0(x)−1
).
3 If we have the functions ψ0, . . . , ψk ∈ C∞
(Ω, K) already constructed the next one
ψk+1 is multiplied by γ0 and depends only on ψ0, . . . , ψk and on the γ’s. Indeed,
0 = γ∗Ωψ
k+1
=
k+1
X
l, p, q = 0
l + p + q = k + 1
Cl(γp, ψq) = γ0ψk+1+Fk+1(ψ0, . . . , ψk, γ0, . . . , γk+1)
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

88. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
1 For g ∈ S ⊂ R and γ = η(g) we try to construct ψ ∈ RΩ; γ ∗Ω ψ = 1.
2 We work order by order. For k = 0 it’s easy (x 7→ ψ0(x) = γ0(x)−1
).
3 If we have the functions ψ0, . . . , ψk ∈ C∞
(Ω, K) already constructed the next one
ψk+1 is multiplied by γ0 and depends only on ψ0, . . . , ψk and on the γ’s. Indeed,
0 = γ∗Ωψ
k+1
=
k+1
X
l, p, q = 0
l + p + q = k + 1
Cl(γp, ψq) = γ0ψk+1+Fk+1(ψ0, . . . , ψk, γ0, . . . , γk+1)
4 Same construction for left inverse. (Associativity =⇒: right inverse = left inverse).
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

89. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
We follow steps of Lemma 6.1, p.113, in
J. C. Tougeron(1972) book to prove:
21/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

90. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
We follow steps of Lemma 6.1, p.113, in
J. C. Tougeron(1972) book to prove:
(b) Every ϕ ∈ RΩ is equal to
η(f) ∗Ω η(g)∗Ω−1 for some f ∈ R
and g ∈ S.
21/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

91. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
We follow steps of Lemma 6.1, p.113, in
J. C. Tougeron(1972) book to prove:
(b) Every ϕ ∈ RΩ is equal to
η(f) ∗Ω η(g)∗Ω−1 for some f ∈ R
and g ∈ S.
(c) The kernel of η is equal to the
space of functions f ∈ R such that
there is g ∈ S with f ∗ g = 0.
Tougeron’s Lemma: Let Ω be an open set
of Rn, and (ϕi)i∈N a sequence of smooth
functions Ω → K. Then there is a smooth
function α : Rn → R s. t.
1 α takes only values between 0 and 1.
Moreover α(x) = 0 for all x ̸∈ Ω, and
α(x) 0 for all x ∈ Ω.
2 For each nonnegative integer i the
function ϕ′
i : Rn → K defined by
ϕ′
i(x) :=
ϕi(x)α(x) if x ∈ Ω
0 if x ̸∈ Ω
is
smooth.
21/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

92. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(Returning to the) Sketch of the proof
(b) Every ϕ ∈ RΩ is equal to η(f) ∗Ω η(g)∗Ω−1 for some f ∈ R and g ∈ S.
22/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

93. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(Returning to the) Sketch of the proof
(b) Every ϕ ∈ RΩ is equal to η(f) ∗Ω η(g)∗Ω−1 for some f ∈ R and g ∈ S.
To prove that we need some ingredients:
- For a compact set K and non negative integer m define:
pK,m(f) = max{|Dn
f(v)| | n ≤ m, τX(v) ∈ K and h(v, v) ≤ 1}.
- Where pK,m : A → R
- Which will define an exhaustive system of seminorms, hence a locally convex
topological vector space which is known to be metric and sequentially complete,
hence Fréchet.
22/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

94. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

95. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

96. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

97. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
b. (gj)j∈N C∞
-functions X → R with gj|Kj = 1 and supp(gj) ⊂ Kj+1,
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

98. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
b. (gj)j∈N C∞
-functions X → R with gj|Kj = 1 and supp(gj) ⊂ Kj+1,
c. For all j ∈ N: ϵjpKj+1,j(gj) 1
2
j
,
d. For all i ≤ j ∈ N: ϵj
Pi
k=0 pKj+1,j (Ck|Ω(ϕi−k, η(gj))) 1
2
j
.
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

99. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
b. (gj)j∈N C∞
-functions X → R with gj|Kj = 1 and supp(gj) ⊂ Kj+1,
c. For all j ∈ N: ϵjpKj+1,j(gj) 1
2
j
,
d. For all i ≤ j ∈ N: ϵj
Pi
k=0 pKj+1,j (Ck|Ω(ϕi−k, η(gj))) 1
2
j
.
3 Then g(N) =
PN
j=0 ϵjgj → g (N → ∞) s.t. ∀ x ∈ Ω : g(x) 0 and g|XΩ = 0,
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

100. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
b. (gj)j∈N C∞
-functions X → R with gj|Kj = 1 and supp(gj) ⊂ Kj+1,
c. For all j ∈ N: ϵjpKj+1,j(gj) 1
2
j
,
d. For all i ≤ j ∈ N: ϵj
Pi
k=0 pKj+1,j (Ck|Ω(ϕi−k, η(gj))) 1
2
j
.
3 Then g(N) =
PN
j=0 ϵjgj → g (N → ∞) s.t. ∀ x ∈ Ω : g(x) 0 and g|XΩ = 0,
4 and for each i ∈ N: the sequence of unique fiN : X → K such that
η(fiN ) =
Pi
k=0 Ck|Ω(ϕi−k, η(g(N))) and fiN |XΩ = 0 converges to a smooth
fi : X → C with η(fi) =
Pi
k=0 Ck|Ω(ϕi−k, η(g)) solving the problem.
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

101. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(c) The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S
with f ∗ g = 0.
24/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

102. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(c) The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S
with f ∗ g = 0.
1 Clearly {f ∈ R; f ⋆Ω g = 0, g ∈ S} ⊂ ker(η) because f ∗ g = 0 ⇒ η(f) ∗Ω η(g) = 0
⇒ η(f) = 0 since η(g) is invertible in RΩ.
24/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

103. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(c) The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S
with f ∗ g = 0.
1 Clearly {f ∈ R; f ⋆Ω g = 0, g ∈ S} ⊂ ker(η) because f ∗ g = 0 ⇒ η(f) ∗Ω η(g) = 0
⇒ η(f) = 0 since η(g) is invertible in RΩ.
2 Conversely, f ∈ ker(η) ⇒ fi(x) = 0, ∀x ∈ Ω.
24/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

104. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(c) The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S
with f ∗ g = 0.
1 Clearly {f ∈ R; f ⋆Ω g = 0, g ∈ S} ⊂ ker(η) because f ∗ g = 0 ⇒ η(f) ∗Ω η(g) = 0
⇒ η(f) = 0 since η(g) is invertible in RΩ.
2 Conversely, f ∈ ker(η) ⇒ fi(x) = 0, ∀x ∈ Ω.
3 Taking g as in the property (b)(fonction aplatisseur), for ϕ0 = 1, ϕi = 0 for i ≥ 1 we
obtain
∀x ∈ X, (f ⋆ g)i = 0.
24/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

105. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

106. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
For open sets U ⊃ V denote by
ηU
V : RU → RV be the restriction
morphism (ηU for ηX
U ).
The family RU
U∈X
with the restriction
morphisms ηU
V defines a sheaf of
K-algebras over X.
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

107. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
For open sets U ⊃ V denote by
ηU
V : RU → RV be the restriction
morphism (ηU for ηX
U ).
The family RU
U∈X
with the restriction
morphisms ηU
V defines a sheaf of
K-algebras over X.
Let x0 ∈ X and Xx0
⊂ X the set of all open
sets containing x0.
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

108. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
For open sets U ⊃ V denote by
ηU
V : RU → RV be the restriction
morphism (ηU for ηX
U ).
The family RU
U∈X
with the restriction
morphisms ηU
V defines a sheaf of
K-algebras over X.
Let x0 ∈ X and Xx0
⊂ X the set of all open
sets containing x0.
Considering
R̃x0 = {(U, f); U ∈ Xx0 , f ∈ C∞
(U, K)[[λ]]}
The Stalk at x0 is
Rx0 =
R̃x0
∼
=
S
U∈Xx0
RU
!
∼
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

109. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
For open sets U ⊃ V denote by
ηU
V : RU → RV be the restriction
morphism (ηU for ηX
U ).
The family RU
U∈X
with the restriction
morphisms ηU
V defines a sheaf of
K-algebras over X.
Let x0 ∈ X and Xx0
⊂ X the set of all open
sets containing x0.
Considering
R̃x0 = {(U, f); U ∈ Xx0 , f ∈ C∞
(U, K)[[λ]]}
The Stalk at x0 is
Rx0 =
R̃x0
∼
=
S
U∈Xx0
RU
!
∼
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

110. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

111. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

112. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

113. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
Consider S = S(x0) = {g ∈ R | g0(x0) ̸= 0}
and
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

114. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
Consider S = S(x0) = {g ∈ R | g0(x0) ̸= 0}
and
I = Ix0 = {g ∈ R | g0(x0) = 0}
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

115. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
Consider S = S(x0) = {g ∈ R | g0(x0) ̸= 0}
and
I = Ix0 = {g ∈ R | g0(x0) = 0}
S = R I is a multiplicative subset and Ix0
maximal ideal of R.
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

116. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
Consider S = S(x0) = {g ∈ R | g0(x0) ̸= 0}
and
I = Ix0 = {g ∈ R | g0(x0) = 0}
S = R I is a multiplicative subset and Ix0
maximal ideal of R.
Finally we present the same result as before for
germs:
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

117. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Theorem: Using the previously fixed notations we get for any point x0 ∈ X:
1 (Rx0 , ∗x0 ) together with the morphism ηx0 : R → Rx0 consitutes a right
K-algebra of fractions for (R, S(x0)).
2 As an immediate consequence we have that S(x0) is a right denominator set.
3 This implies in particular that the algebraic localization RS−1 of R with
respect to S = S(x0) is isomorphic to the concrete stalk Rx0 as unital
K-algebras.
27/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

118. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Example
The following example provides a non-Ore subset which is a subset of an Ore
subset.
28/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

119. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Example
The following example provides a non-Ore subset which is a subset of an Ore
subset. Consider C∞(R2, R) with the standard star product ∗ given by the formula
f ∗ g =
∞
X
k=0
λk
k!
∂kf
∂pk
∂kg
∂xk
.
Let R = C∞(R2, R)[[λ]], and let Ω ⊂ R2 be the open set of all (x, p) ∈ R2 where
p ̸= 0. Then,
The subset S = {1, p, p2, p3, . . .} ⊂ R is a multiplicative subset of (R, ∗) which is
contained in the Ore subset SΩ but which is neither right nor left Ore.
For instance, for r = (x, p) 7→ ex and s = (x, p) 7→ p we can not find r′, s′ such
that, r′ ∗ s = s′ ∗ r
28/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

120. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Question: Localization commutes with deformation?
Proposition: Let A be a commutative
unital K-algebra and a differential star
product ∗ =
P∞
i=0 λiCi on R := A[[λ]].
For any multiplicative subset S0 ⊂ A there
exists a unique star product ∗S0 on
AS0 [[λ]] such that the numerator map η
canonically extended as a K[[λ]]-linear map
(also denoted η) A[[λ]] → AS0 [[λ]] is a
morphism of unital K[[λ]]-algebras.
29/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

121. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Question: Localization commutes with deformation?
Proposition: Let A be a commutative
unital K-algebra and a differential star
product ∗ =
P∞
i=0 λiCi on R := A[[λ]].
For any multiplicative subset S0 ⊂ A there
exists a unique star product ∗S0 on
AS0 [[λ]] such that the numerator map η
canonically extended as a K[[λ]]-linear map
(also denoted η) A[[λ]] → AS0 [[λ]] is a
morphism of unital K[[λ]]-algebras.
With the above structures A, S0, ∗ consider
the subset S = S0 + λR ⊂ R = A[[λ]].
The subset S = S0 + λR is a multiplicative
subset of the algebra (R, ∗)
Moreover, its image under η consists of
invertible elements of the K[[λ]]-algebra
AS0 [[λ]], ∗S0
.
It follows that there is a canonical
morphism
Φ :
A[[λ]]
S
∗S
→ AS0 [[λ]], ∗S0
.
29/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

122. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Thanks for your attention!!!
30/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

123. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Thanks for your attention!!!
Danke für Ihre Aufmerksamkeit!!!
Obrigado pela sua atenção!!!
Merci de votre attention!!!
30/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

124. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Bibliography
Araujo, H., Bordemann, M., Hurle, B.: Noncommutative localization in smooth
deformation quantization. Preprint, ArXiv:2010.15701 2020.
Bayen, F., Flato, M., Frønsdal, C., Licherowicz, A., Sternheimer, D.: Deformation
theory and quantization. I, II. Annals of Phys. 111, 61-110, 111-151 (1978).
Lam, T.Y.: Lectures on Modules and Rings. Springer Verlag, Berlin, 1999.
Mac Lane, S.: Categories for the Working Mathematician. 2nd ed., Springer, New
York, 1998.
Škoda, Z.: Noncommutative localization in noncommutative geometry,
arXiv:math/0403276v2, 2005.
Tougeron, J.-C.: Idéaux des fonctions différentiables, Springer Verlag, Berlin, 1972.
31/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization