Test

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 Universit´ e de Haute-Alsace IRIMAS - Depart´ ement de Mathematiques Arbeitsgruppenseminar Analysis Universit¨ at Potsdam, Deutschland, 04 dez 2020 1/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

products Deformation quantization Localization In this work we talk about: 2/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products Deformation quantization Localization In this work we talk about: Algebraic localization Analytic localization 2/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products Deformation quantization Localization In this work we talk about: Algebraic localization Analytic localization Compare this two types of localization. 2/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

follows: Localization 3/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

follows: Localization - Commutative and noncommutative cases 3/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

follows: Localization - Commutative and noncommutative cases - Ore conditions 3/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

follows: Localization - Commutative and noncommutative cases - Ore conditions Deformation quantization 3/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

follows: Localization - Commutative and noncommutative cases - Ore conditions Deformation quantization - Star products - Concrete localization 3/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

follows: Localization - Commutative and noncommutative cases - Ore conditions Deformation quantization - Star products - Concrete localization Results 3/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

follows: Localization - Commutative and noncommutative cases - Ore conditions Deformation quantization - Star products - Concrete localization Results - and comments 3/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 → x 1 and the elements of R \ {0R} have become invertible in R, x 1 −1 = 1 x . 4/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 → x 1 and the elements of R \ {0R} have become invertible in R, x 1 −1 = 1 x . The morphism x → x 1 is in general not injective. 4/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

Ore conditions Conventions and definitions From now on -if we have not specified it before- let us consider: 5/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

Ore conditions Localization of a commutative K-algebra 6/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 → r 1 . This map defines a K-algebra stucture of RS. 6/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

Ore conditions Localization of a commutative K-algebra The construction of the K-algebra RS gives us important properties. 7/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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) L − − − − − − − → RS KAlgMS KAlg (R, U(R)) ← − − − − − − U R 8/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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) L − − − − − − − → RS KAlgMS KAlg (R, U(R)) ← − − − − − − U R Proposition: The functor L also exists in the noncommutative case and L is left adjoint to U. 8/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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.: ˇ RS with ˇ η(R,S) = ˇ η : R → ˇ RS is said to be a right K-algebra of fractions of (R, S) if: a. ˇ η(R,S) is S-inverting, b. Every element of ˇ RS is of the form ˇ η(r) ˇ η(s) −1 for r ∈ R and s ∈ S; c. ker(ˇ η) = {r ∈ R | rs = 0, for some s ∈ S} =: I(R,S) =: I. 10/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

Ore conditions Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true: 11/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

Ore conditions Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true: 1 ˇ RS is a right K-algebra of fractions ⇐⇒ S is a right denominator set. 11/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

Ore conditions Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true: 1 ˇ RS is a right K-algebra of fractions ⇐⇒ S is a right denominator set. 2 If this is the case each such pair ( ˇ RS, ˇ η) is universal and then each ˇ RS is isomorphic to the canonical localized algebra RS. 3 Each ˇ RS 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

Ore conditions Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true: 1 ˇ RS is a right K-algebra of fractions ⇐⇒ S is a right denominator set. 2 If this is the case each such pair ( ˇ RS, ˇ η) is universal and then each ˇ RS is isomorphic to the canonical localized algebra RS. 3 Each ˇ RS 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

Ore conditions Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true: 1 ˇ RS is a right K-algebra of fractions ⇐⇒ S is a right denominator set. 2 If this is the case each such pair ( ˇ RS, ˇ η) is universal and then each ˇ RS is isomorphic to the canonical localized algebra RS. 3 Each ˇ RS 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

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

products on open sets For a K-vector space V let V [[λ]] = {v = ∞ 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products on open sets For a K-vector space V let V [[λ]] = {v = ∞ 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 = ∞ k=0 Ck(f, g)λk, where Ck : C∞(X) ⊗ C∞(X) → C∞(X) are bidifferential operators. 13/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products on open sets For a K-vector space V let V [[λ]] = {v = ∞ 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 = ∞ 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 = ∞ k=0 λk k! ∂kf ∂pk ∂kg ∂xk (Multiplication of diff. operators) 13/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products on open sets Localization on open sets 15/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products on open sets Question: How to relate localization with star-products? 16/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

products on open sets Question: How to relate localization with star-products? Let ∗ = ∞ 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products on open sets Question: How to relate localization with star-products? Let ∗ = ∞ 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 = ∞ 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products on open sets Question: How to relate localization with star-products? Let ∗ = ∞ 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 = ∞ 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 → f|Ω 16/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

products on open sets Abstract localization Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} . 17/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

open sets Germs A non Ore example One of the results of my thesis is the following: 19/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

open sets Germs A non Ore example Sketch of the proof: (a) η is S- inverting. 20/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 → ψ0 (x) = γ0 (x)−1). 20/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 → 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∗Ω ψ k+1 = k+1 l, p, q = 0 l + p + q = k + 1 Cl (γp , ψq ) = γ0 ψk+1 +Fk+1 (ψ0 , . . . , ψk , γ0 , . . . , γk+1 ) 20/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 → 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∗Ω ψ k+1 = k+1 l, p, q = 0 l + p + q = k + 1 Cl (γp , ψq ) = γ0 ψk+1 +Fk+1 (ψ0 , . . . , ψk , γ0 , . . . , γk+1 ) 4 Same construction for left inverse. (Associativity =⇒: right inverse = left inverse). 20/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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{|Dnf(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 Fr´ echet. 22/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

open sets Germs A non Ore example Sketch of the proof 1 Consider the equation η(f) = ϕ ∗Ω η(g) = ∞ i=0 λi i k=0 Ck|Ω(ϕi−k, η(g)). 23/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

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

open sets Germs A non Ore example Sketch of the proof 1 Consider the equation η(f) = ϕ ∗Ω η(g) = ∞ i=0 λi i k=0 Ck|Ω(ϕi−k, η(g)). 2 Ansatz: g (no λs!), g ≥ 0, and g = ∞ j=0 ϵjgj where ϵj > 0 and a. (Kn )n∈N sequence of compact subsets with Kj ⊂ Kj+1 and 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

open sets Germs A non Ore example Sketch of the proof 1 Consider the equation η(f) = ϕ ∗Ω η(g) = ∞ i=0 λi i k=0 Ck|Ω(ϕi−k, η(g)). 2 Ansatz: g (no λs!), g ≥ 0, and g = ∞ j=0 ϵjgj where ϵj > 0 and a. (Kn )n∈N sequence of compact subsets with Kj ⊂ Kj+1 and 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: ϵj pKj+1,j (gj ) < 1 2 j , d. For all i ≤ j ∈ N: ϵj i k=0 pKj+1,j (Ck |Ω (ϕi−k , η(gj ))) < 1 2 j . 23/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

open sets Germs A non Ore example Sketch of the proof 1 Consider the equation η(f) = ϕ ∗Ω η(g) = ∞ i=0 λi i k=0 Ck|Ω(ϕi−k, η(g)). 2 Ansatz: g (no λs!), g ≥ 0, and g = ∞ j=0 ϵjgj where ϵj > 0 and a. (Kn )n∈N sequence of compact subsets with Kj ⊂ Kj+1 and 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: ϵj pKj+1,j (gj ) < 1 2 j , d. For all i ≤ j ∈ N: ϵj i k=0 pKj+1,j (Ck |Ω (ϕi−k , η(gj ))) < 1 2 j . 3 Then g(N) = N j=0 ϵjgj → g (N → ∞) s.t. ∀ x ∈ Ω : g(x) > 0 and g|X\Ω = 0, 23/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ˜ Rx0 = {(U, f); U ∈ Xx0 , f ∈ C∞(U, K)[[λ]]} The Stalk at x0 is Rx0 = ˜ Rx0 ∼ = U∈Xx0 RU ∼ 25/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ˜ Rx0 / ∼x0 and by ηU x0 : RU → Rx0 the restriction of the canonical projection ˜ Rx0 → Rx0 to RU ⊂ ˜ Rx0 . 26/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ˜ Rx0 / ∼x0 and by ηU x0 : RU → Rx0 the restriction of the canonical projection ˜ Rx0 → Rx0 to RU ⊂ ˜ Rx0 . We get a unital associative K-algebra denoted by Rx0 , ∗x0 26/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ˜ Rx0 / ∼x0 and by ηU x0 : RU → Rx0 the restriction of the canonical projection ˜ Rx0 → Rx0 to RU ⊂ ˜ Rx0 . 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ˜ Rx0 / ∼x0 and by ηU x0 : RU → Rx0 the restriction of the canonical projection ˜ Rx0 → Rx0 to RU ⊂ ˜ Rx0 . 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ˜ Rx0 / ∼x0 and by ηU x0 : RU → Rx0 the restriction of the canonical projection ˜ Rx0 → Rx0 to RU ⊂ ˜ Rx0 . 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ˜ Rx0 / ∼x0 and by ηU x0 : RU → Rx0 the restriction of the canonical projection ˜ Rx0 → Rx0 to RU ⊂ ˜ Rx0 . 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 = ∞ 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) → ex and s = (x, p) → p we can not find r′, s′ such that, r′ ∗ s = s′ ∗ r 28/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ∗ = ∞ 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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 ∗ = ∞ 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 - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

open sets Germs A non Ore example Thanks for your attention!!! 30/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

open sets Germs A non Ore example Thanks for your attention!!! Danke f¨ ur Ihre Aufmerksamkeit!!! Obrigado pela sua aten¸ c˜ ao!!! Merci de votre attention!!! 30/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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. ˇ Skoda, Z.: Noncommutative localization in noncommutative geometry, arXiv:math/0403276v2, 2005. Tougeron, J.-C.: Id´ eaux des fonctions diff´ erentiables, Springer Verlag, Berlin, 1972. 31/31 Hamilton ARAUJO - Universit´ e de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization

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