A survey on simple derivations and their isotropy groupseventoribenboim/DanielLevcovitz.pdf ·...

d-simplicityThe result of Shamsuddin

The isotropy group of a derivationsDerivations of the polynomial ring K [X , Y ]

References

A survey on simple derivations and theirisotropy groups

Daniel Levcovitz

Universidade de São Paulo, USP

e-mail: lev@icmc.usp.br

90th Ribenboim Birthday, October, 2018.

Daniel Levcovitz

References

Summary

1 d-simplicityApplications to Commutative Algebra

2 The result of Shamsuddin

3 The isotropy group of a derivationsSimple Shamsuddin derivations

4 Derivations of the polynomial ring K [X ,Y ]

5 References

Daniel Levcovitz

References

Applications to Commutative Algebra

d-simplicity

Let K be a field of characteristic zero A be a K−algebra and da K -derivation of A.(d(a + b) = d(a) + d(b); d(ab) = d(a)b + ad(b); d(K ) = 0).

An ideal I of A is a d-ideal (or d-stable ) if d(I) ⊆ I. The algebraA is d-simple if (0) e A are the only d-ideals of A.

Daniel Levcovitz

References

d-simplicity

Daniel Levcovitz

References

d-simplicity

Daniel Levcovitz

References

D-simplicity

Let ∅ 6= D ⊂ Der(A) be a family (finite or not) of derivations deA. The ring A is D-simple (or D-stable) if it is d-simple for allderivation d ∈ D. Finally, A is differentially simple when it isD-simple for D = Der (A).

Daniel Levcovitz

References

D-simplicity

Daniel Levcovitz

References

D-simplicity

Daniel Levcovitz

References

Example 1

A := K [X ]

d =∂

K [X ] is ∂∂X - simple.

Daniel Levcovitz

References

Example 1

A := K [X ]

d =∂

K [X ] is ∂∂X - simple.

Daniel Levcovitz

References

Example 2

A := K [X ]

d ′ = X∂

∂XK[X] is NOT d ′-simple.In fact, (X ) is a d ′-ideal, since

d ′(X ) = X∂

∂X(X ) = X ∈ (X ).

Daniel Levcovitz

References

Example 2

A := K [X ]

d ′ = X∂

d ′(X ) = X∂

∂X(X ) = X ∈ (X ).

Daniel Levcovitz

References

Example 2

A := K [X ]

d ′ = X∂

d ′(X ) = X∂

∂X(X ) = X ∈ (X ).

Daniel Levcovitz

References

Example 3

A := K [X ,Y ]

d =∂

∂XK [X ,Y ] is NOT d-simple.

(Y ) is a d-ideal.

Daniel Levcovitz

References

Example 3

A := K [X ,Y ]

d =∂

(Y ) is a d-ideal.

Daniel Levcovitz

References

Example 3

A := K [X ,Y ]

d =∂

(Y ) is a d-ideal.

Daniel Levcovitz

References

Example 4 (Bergman e Lequain)

A := K [X ,Y ]

d =∂

∂X+ (1 + XY )

∂YK [X ,Y ] is d-simple.

Daniel Levcovitz

References

Example 4 (Bergman e Lequain)

A := K [X ,Y ]

d =∂

∂X+ (1 + XY )

∂YK [X ,Y ] is d-simple.

Daniel Levcovitz

References

The result of Seidenberg

Let A := K [x1, . . . , xn] be an affine domain.

Theorem(Seidenberg, 1967)A is differentially simple⇔ A is regular.

BUT, Hart (1975) showed that when K [x1, · · · , xn] isdifferentially simple, it may not exist a single derivation d suchthat k [x1, · · · , xn] is d-simple.For example

A :=Q[X ,Y ,Z ]

(X 2 + Y 2 + Z 2 − 1)

Daniel Levcovitz

References

A :=Q[X ,Y ,Z ]

(X 2 + Y 2 + Z 2 − 1)

Daniel Levcovitz

References

A :=Q[X ,Y ,Z ]

(X 2 + Y 2 + Z 2 − 1)

Daniel Levcovitz

References

The result of Hart

Let A := S−1(K [x1, . . . , xn]) be a domain essentially of finitetype.

Theorem(Hart, 1975)A admits a simple derivation d ⇔ A is regular.

Daniel Levcovitz

References

The result of Hart

Let A := S−1(K [x1, . . . , xn]) be a domain essentially of finitetype.

Theorem(Hart, 1975)A admits a simple derivation d ⇔ A is regular.

Daniel Levcovitz

References

The result of Shamsuddin

Let A be any commutative ring that contains Q and let d be aderivation of A. Given any polynomial f (Y ) ∈ A[Y ], we canextend d to a derivation of A[Y ] putting:

d(Y ) := f (Y ).

We are interested when f (Y ) has degree one, that is:

d(Y ) = aY + b, a,b ∈ A, a 6= 0.

Daniel Levcovitz

References

Let A be any commutative ring that contains Q and let d be aderivation of A. Given any polynomial f (Y ) ∈ A[Y ], we canextend d to a derivation of A[Y ] putting:

d(Y ) := f (Y ).

We are interested when f (Y ) has degree one, that is:

d(Y ) = aY + b, a,b ∈ A, a 6= 0.

Daniel Levcovitz

References

Theorem (Shamsuddin, 1977 )Let d be a simple derivation of A. Put

d(Y ) = aY + b,a,b ∈ A.

Then,A[Y ] is d-simple

The diff. equation d(r) = ar + b does not have a solution in A

This differential equation over A is called the ODE associated tothe derivation d .

Daniel Levcovitz

References

d(Y ) = aY + b,a,b ∈ A.

Daniel Levcovitz

References

d(Y ) = aY + b,a,b ∈ A.

Daniel Levcovitz

References

The example of Bergam-Lequain, bis.

A := K [X ]

d =∂

d(Y ) := 1 + XY

A[Y ] = K [X ,Y ]

d =∂

∂X+ (1 + XY )

Daniel Levcovitz

References

A := K [X ]

d =∂

d(Y ) := 1 + XY

A[Y ] = K [X ,Y ]

d =∂

∂X+ (1 + XY )

Daniel Levcovitz

References

A := K [X ]

d =∂

d(Y ) := 1 + XY

A[Y ] = K [X ,Y ]

d =∂

∂X+ (1 + XY )

Daniel Levcovitz

References

A := K [X ]

d =∂

d(Y ) := 1 + XY

A[Y ] = K [X ,Y ]

d =∂

∂X+ (1 + XY )

Daniel Levcovitz

References

The example of Bergamn-Lequain, bis.

Shamsuddin:

k [X ,Y ] is (∂

∂X+ (1 + XY )

∂Y)− simple

The ODE r ′ = Xr + 1 has no solution in k [X ]

Daniel Levcovitz

References

The example of Bergamn-Lequain, bis.

Shamsuddin:

k [X ,Y ] is (∂

∂X+ (1 + XY )

∂Y)− simple

The ODE r ′ = Xr + 1 has no solution in k [X ]

Daniel Levcovitz

References

Other example of simple ´´ Shamsuddin´´ derivations

Coutinho (Collier) (1999):

R := K [X ,Y ,Z ]

d :=∂

∂X+ (a(X )Y + b(X ))

∂Y+ (c(X )Z + d(X ))

Daniel Levcovitz

References

Shamsuddin derivations

A derivations of K [X1, . . . ,Xn] is a Shamsuddin derivation if ithas the following form:

d = ∂1 + (a2(X1)X2 + b2(X1))∂2 + · · ·+ (an(X1)Xn + bn(X1))∂n

whereai(X1),bi(X1) ∈ K [X1], j = 2, . . . ,n.

Daniel Levcovitz

References

Shamsuddin derivations

A derivations of K [X1, . . . ,Xn] is a Shamsuddin derivation if ithas the following form:

whereai(X1),bi(X1) ∈ K [X1], j = 2, . . . ,n.

Daniel Levcovitz

References

Shamsuddin derivations that are NOT simple

Not all Shamsuddin derivations are simple:

Example

K [X ,Y ]

d =∂

∂X+ (XY )

∂YThe ideal (Y) is d-stable.

Example

K [X ,Y ,Z ]

d = ∂X + (XY + 1)∂Y + (XZ + 1)∂Z

The ideal I = (Y − Z ) is d-stable, d(Y − Z ) = X (Y − Z )

Daniel Levcovitz

References

Example

K [X ,Y ]

d =∂

∂X+ (XY )

Example

K [X ,Y ,Z ]

d = ∂X + (XY + 1)∂Y + (XZ + 1)∂Z

Daniel Levcovitz

References

Example

K [X ,Y ]

d =∂

∂X+ (XY )

Example

K [X ,Y ,Z ]

d = ∂X + (XY + 1)∂Y + (XZ + 1)∂Z

Daniel Levcovitz

References

Questions

Question A: When a Shamsuddin derivation is simple?

Question B: Ate there simple derivations that are NOTShamsuddin derivations?

Daniel Levcovitz

References

Questions

Question A: When a Shamsuddin derivation is simple?

Question B: Ate there simple derivations that are NOTShamsuddin derivations?

Daniel Levcovitz

References

A result of Lequain, 2008

TheoremIt is possible do decide (effectively) when a Shamsuddinderivation

is simple or not.One has to compute some invariants of the polynomialsai(X1),bi(X1) ∈ K [X1].

Daniel Levcovitz

References

A result of Lequain, 2008

TheoremIt is possible do decide (effectively) when a Shamsuddinderivation

is simple or not.One has to compute some invariants of the polynomialsai(X1),bi(X1) ∈ K [X1].

Daniel Levcovitz

References

Simple derivations that are not Shamsuddin

Example(Archer, 1981) (based in Shamsuddin)

A = K [X1, . . . ,Xn]

d = ∂1 +n∑

(1 + Xi−1Xi)∂i

Example(Maciejewski, Moulin-Ollagnier, Nowicki, 2001). They studiedwhen a derivation of the form

d = ∂X + (Y 2 − p(X ))∂Y

is simple. For example, if p(X ) has odd degree,then it is simple.Daniel Levcovitz

References

A = K [X1, . . . ,Xn]

d = ∂1 +n∑

(1 + Xi−1Xi)∂i

d = ∂X + (Y 2 − p(X ))∂Y

References

A = K [X1, . . . ,Xn]

d = ∂1 +n∑

(1 + Xi−1Xi)∂i

d = ∂X + (Y 2 − p(X ))∂Y

References

Observation and Problem

Observation: Archer showed that there always exists a base ofthe free module of derivations

DerK (K [X1, . . . ,Xn])

formed by simple derivations.

Totally open problem: Classify simple derivations ofpolynomials rings.

Daniel Levcovitz

References

DerK (K [X1, . . . ,Xn])

Daniel Levcovitz

References

DerK (K [X1, . . . ,Xn])

Daniel Levcovitz

References

Simple Shamsuddin derivations

Definition

Let d be a K -derivation of a K -algebra A. Consider the fullgroup AutK (A) of K -automorphisms of A.The isotropy group of d is the subgroup of AutK (A) defined by

Aut(A)d = {ρ ∈ AutK (A)| ρdρ−1 = d}.

Note that the isotropy group is just de stabilizer subgroup of theaction by conjugation:

AutK (A)× DerK (A)→ DerK (A)

(ρ,d) 7→ ρdρ−1.

Daniel Levcovitz

References

Definition

(ρ,d) 7→ ρdρ−1.

Daniel Levcovitz

References

Definition

(ρ,d) 7→ ρdρ−1.

Daniel Levcovitz

References

Easy example

A = K [X ]

d =∂

∂XAut(K [X ])d = {ρ : X 7→ a + X ,a ∈ K}

This is an infinite group.

Daniel Levcovitz

References

Easy example

A = K [X ]

d =∂

∂XAut(K [X ])d = {ρ : X 7→ a + X ,a ∈ K}

This is an infinite group.

Daniel Levcovitz

References

Baltazar’s thesis

He considered the polynomial ring K [X ,Y ].

Example

A = K [X ,Y ]

d =∂

∂XHe computed:

Aut(K [X ,Y ])d = {ρ : (X ,Y ) 7→ (X + p(Y ),aY + b)},

where a,b ∈ K ,a 6= 0This is an infinite group.

Daniel Levcovitz

References

Baltazar’s thesis

Example

A = K [X ,Y ]

d =∂

∂XHe computed:

Daniel Levcovitz

References

Baltazar’s thesis

Example

A = K [X ,Y ]

d =∂

∂XHe computed:

Daniel Levcovitz

References

Baltazar’s theorem

Theorem(Baltazar, 2014) Let d be a Shamsuddin derivation of K [X ,Y ].Suppose that d is simple. Then its isotropy groupAut(K [X ,Y ])d is trivial.

Then Baltazar and Pan (his supervisor) conjectured:

Conjecture (Baltazar and Pan): If d is a simple derivation ofan affine K -algebra, then its isotropy group is finite.

Not true even for K [X1, . . . ,Xn] if n ≥ 3.

Daniel Levcovitz

References

Baltazar’s theorem

Theorem(Baltazar, 2014) Let d be a Shamsuddin derivation of K [X ,Y ].Suppose that d is simple. Then its isotropy groupAut(K [X ,Y ])d is trivial.

Then Baltazar and Pan (his supervisor) conjectured:

Conjecture (Baltazar and Pan): If d is a simple derivation ofan affine K -algebra, then its isotropy group is finite.

Not true even for K [X1, . . . ,Xn] if n ≥ 3.

Daniel Levcovitz

References

Mendes and Pan theorem

Theorem(Mendes, Pan, 2018) If d be a simple derivation of K [X ,Y ],then its isotropy group Aut(K [X ,Y ])d is trivial.

Daniel Levcovitz

References

Theorem(Bertoncello,—, 2016.) Let d be a simple Shamsuddinderivation of K [X1, . . . ,Xn],n ≥ 2. Then its isotropy groupAut(K [X1, . . . ,Xn])d is trivial.

Remark: The proof of this theorem is heavily based inLequain’s characterization of simple Shamsuddin derivations.

Daniel Levcovitz

References

Theorem(Bertoncello,—, 2016.) Let d be a simple Shamsuddinderivation of K [X1, . . . ,Xn],n ≥ 2. Then its isotropy groupAut(K [X1, . . . ,Xn])d is trivial.

Remark: The proof of this theorem is heavily based inLequain’s characterization of simple Shamsuddin derivations.

Daniel Levcovitz

References

A new conjecture

Conjecture (Baltazar, Bertoncello,–, Mendes, Pan): Let d bea Shamsuddin derivation of the polynomial ring

K [X1, . . . ,Xn],n ≥ 2.

Then d is simple if, and only if, its isotropy group is trivial.

Daniel Levcovitz

References

Definition

Let K [X ,Y ] be the polynomial ring in two variables over a fieldK of characteristic zero. A derivation d of K [X ,Y ] hasY -degree n if d(X ) = 1 and d(Y ) has degree n as a polynomialin Y with coefficients in K [X ].

Notation: d(Y ) = hnY n + hn−1Y n−1 + · · ·+ h1Y + h0.Then d = ∂

∂X + (hnY n + hn−1Y n−1 + · · ·+ h1Y + h0)∂∂Y , where

hi(X ) ∈ K [X ].

Daniel Levcovitz

References

Definition

hi(X ) ∈ K [X ].

Daniel Levcovitz

References

Definition

hi(X ) ∈ K [X ].

Daniel Levcovitz

References

Definition

hi(X ) ∈ K [X ].

Daniel Levcovitz

References

Definition

A quadratic derivation has Y -degree 2:d = ∂

∂X + (h2Y 2 + h1Y + h0)∂∂Y .

A cubic derivation has Y -degree 3:d = ∂

∂X + (h3Y 3 + h2Y 2 + h1Y + h0)∂∂Y .

Daniel Levcovitz

References

Definition

A quadratic derivation has Y -degree 2:d = ∂

∂X + (h2Y 2 + h1Y + h0)∂∂Y .

A cubic derivation has Y -degree 3:d = ∂

∂X + (h3Y 3 + h2Y 2 + h1Y + h0)∂∂Y .

Daniel Levcovitz

References

Theorem

Theorem(Bertoncello,—, 2016.) Let d be a derivation of the polynomialring in two variables K [X ,Y ] of Y -degree n ≥ 2. Letρ ∈ (K [X ,Y ])d be in the isotropy group of d. Then,

(i) ρ(X ) = X + α, (α ∈ K ) and ρ(Y ) = b0 + b1Y withb0 ∈ K [X ],b1 ∈ K ? and satisfies bn−1

1 = 1.(ii) If hn(X ) ∈ K [X ] \ K and b1 = 1, then b0 = 0. In this case

ρ =id.(iii) If If hn(X ) ∈ K [X ] \ K , h0 6= 0 and b0 = 0, then b1 = 1. In

this case ρ =id.(iv) If If hn(X ) ∈ K [X ] \ K , d is simple and b0 = 0, then b1 = 1.

In this case ρ =id.

Daniel Levcovitz

References

Theorem

Daniel Levcovitz

References

Theorem

Daniel Levcovitz

References

Theorem

Daniel Levcovitz

References

Corollaries

CorollaryLet d be a derivation in two variables of Y -degree n ≥ 2 withhn(X ) ∈ K [X ] \ K . Let µn−1(K ) denote the cyclic group of n − 1roots of unity in K . Then

Aut(K [X ,Y ])d ↪→ µn−1(K ).

In particular it is a finite cyclic group.

Proof.Consider the map ϕ : K [X ,Y ]d → µn−1(K ) given by ϕ(ρ) = b1where ρ(Y ) = b0 + b1Y . It is a group homomorphisms. By (i) ofthe theorem above, it is well defined; by (ii) it is injective. Thenthe result follows.

Daniel Levcovitz

References

Corollaries

CorollaryLet d be a derivation in two variables of Y -degree n ≥ 2 withhn(X ) ∈ K [X ] \ K . Let µn−1(K ) denote the cyclic group of n − 1roots of unity in K . Then

Aut(K [X ,Y ])d ↪→ µn−1(K ).

In particular it is a finite cyclic group.

Proof.Consider the map ϕ : K [X ,Y ]d → µn−1(K ) given by ϕ(ρ) = b1where ρ(Y ) = b0 + b1Y . It is a group homomorphisms. By (i) ofthe theorem above, it is well defined; by (ii) it is injective. Thenthe result follows.

Daniel Levcovitz

References

Corollaries

Corollary(i) If d is a quadratic derivation in two variables with

h2(X ) ∈ K [X ] \ K , then its isotropy group is trivial.(ii) If d is a cubic derivation in two variables with

h3(X ) ∈ K [X ] \ K , then its isotropy group is either trivial ora group of order 2 (and both cases occur).

Daniel Levcovitz

References

Corollaries

Corollary(i) If d is a quadratic derivation in two variables with

h2(X ) ∈ K [X ] \ K , then its isotropy group is trivial.(ii) If d is a cubic derivation in two variables with

h3(X ) ∈ K [X ] \ K , then its isotropy group is either trivial ora group of order 2 (and both cases occur).

Daniel Levcovitz

References

Examples

ExampleLet d be a cubic derivation in two variables given by:

d =∂

∂X+ (h1Y + h3Y 3)

Letρ : X 7→ X ,Y 7→ −Y

Then,K [X ,Y ]d = {id, ρ}

Daniel Levcovitz

References

Examples

d =∂

∂X+ (h1Y + h3Y 3)

Letρ : X 7→ X ,Y 7→ −Y

Daniel Levcovitz

References

Examples

d =∂

∂X+ (h1Y + h3Y 3)

Letρ : X 7→ X ,Y 7→ −Y

Daniel Levcovitz

References

ExampleLet d be the derivation in two variables given by:

d =∂

∂X+ (Y n + pX )

where n ≥ 2,p ∈ K ?.Novicki proved that d is simple.Its isotropy group is trivial.

Daniel Levcovitz

References

d =∂

∂X+ (Y n + pX )

Daniel Levcovitz

References

d =∂

∂X+ (Y n + pX )

Daniel Levcovitz

References

References I

R.Baltazar, On simple Shamsuddin derivations in twovariables, Anais da Academia Brasileira de Ciências, toappear.

R.Baltazar, Tese de Doutorado.

L.N.Bertoncello and D.Levcovitz, On the isotropy group of asimple derivation, preprint.

R. Hart, Derivations on regular local rings of finitelygenerated type, J. London Math. Soc. 10(1975), 292–294.

Y.Lequain, Simple Shamsuddin derivations ofK [X ;Y1, . . . ,Yn], J. Pure Appl. Algebra 212(2008),801–807.

Daniel Levcovitz

References

References II

L.G.Mendes and I.Pan, On plane polynomialautomorphisms commuting with simple derivations,available from Arxiv:1604.04933.

A. Seidenberg, Differential ideals in rings of finitelygenerated type, Amer. J. Math. 89(1967), 22–42.

A. Shamsuddin, Ph.D. Thesis, University of Leeds (1977).

Daniel Levcovitz

A survey on simple derivations and their isotropy groupseventoribenboim/DanielLevcovitz.pdf ·...

Documents

Transcript of A survey on simple derivations and their isotropy groupseventoribenboim/DanielLevcovitz.pdf ·...

Isotropy in oxides - Uni Trier

The return to isotropy of homogeneous turbulence

OUTER DERIVATIONS OF LIE ALGEBRAS · 1967] OUTER DERIVATIONS OF LIE ALGEBRAS 267 outer derivations is a linear sum of the outer derivations, which are obtained as in the first part

Physics Formulae and derivations

Isotropy and Diffuseness in Room Acoustics: Paper ICA2016- 556

Review of gravity model derivations

Prbld Thesis Derivations

يحمرلا يداش DERIVATIONS يحمرلا يداش

Design of Surgical Robots - Page d'accueil / Lirmm.fr / - lirmm · · 2009-09-25Design of Surgical Robots Blake Hannaford ... Isotropy versus azimuth and elevation Points with Isotropy>

ENCH 501 Derivations Nick Dalgliesh

On the hypothesis of isotropy - LMU

An Auxin-Mediated Shift toward Growth Isotropy Promotes ...

physics12-derivations-cp3 - SBG STUDY 12... · Title: physics12-derivations-cp3 Author: CamScanner Subject: physics12-derivations-cp3

Variational Mesh Adaptation: Isotropy and Equidistributionpeople.ku.edu/~wzhuang/research/paper/H01b.pdfVariational Mesh Adaptation: Isotropy and Equidistribution Weizhang Huang Department

physics12-derivations-electrostaticssbgstudy.com/documents/notes/class 12/physics/derivations... · 2019-11-23 · Title: physics12-derivations-electrostatics Author: CamScanner Subject:

ISOTROPY AND AN ASPHALTIC CONCRETE

Intermittency, local isotropy, and non-Gaussian statistics in

Parsing: Derivations, Ambiguity, Precedence, Associativity

Phonology II: derivations, rules, phonotacticspeople.cs.uchicago.edu/...generative-phonology.pdf · Phonology II: derivations, rules, phonotactics ... Generative phonology challenged

Isotropy, Reciprocity and the Generalized Bas-Relief Ambiguityquan/publications/tan-GBR-cvpr07.pdf · Isotropy, Reciprocity and the Generalized Bas-Relief Ambiguity Ping Tan1∗ Satya