combinatorial and additive number theory 2016 [3ex] =1 ...combinatorial and additive number theory...
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Combinatorial and Additive Number
Theory 2016
Additive combinatorics methods in
associative algebras
Vincent Beck
Université d'Orléans
7 janvier 2016
Credits
Thanks to the organizers
Joint work with Cédric Lecouvey of Université de Tours
Preprint : arXiv:1504.02287
Supported by ANR-12-JS01-0003-01 ACORT grant
Credits
Thanks to the organizers
Joint work with Cédric Lecouvey of Université de Tours
Preprint : arXiv:1504.02287
Supported by ANR-12-JS01-0003-01 ACORT grant
Kneser's and Diderrich's Theorems
Theorem Let G be a group and A,B �nite non empty subsets
of G . Assume A is a commutative subset of G (that is to say
aa′ = a′a for every a, a′ in A) and let H := {g ∈ G , gAB = AB}.We have
|AB| > |A|+ |B| − |H|.
Main idea of the proof
Dyson e-transform : �x e ∈ B and let A′ = A ∩ Be−1 and
B ′ = Ae ∪ B . We have A′B ′ ⊂ AB and |A′|+ |B ′| = |A|+ |B|.
Motivations
Is it possible to replace groups by more complex algebraic
structures (�elds, associative algebras over a �eld) ?
Group→→→ �eld, algebra
Combinatorics→→→ linear algebra
In the rest of the talk, k will be an in�nite �eld
Motivations
Is it possible to replace groups by more complex algebraic
structures (�elds, associative algebras over a �eld) ?
Group→→→ �eld, algebra
Combinatorics→→→ linear algebra
In the rest of the talk, k will be an in�nite �eld
Motivations
Is it possible to replace groups by more complex algebraic
structures (�elds, associative algebras over a �eld) ?
Group→→→ �eld, algebra
Combinatorics→→→ linear algebra
In the rest of the talk, k will be an in�nite �eld
Motivations
Is it possible to replace groups by more complex algebraic
structures (�elds, associative algebras over a �eld) ?
Group→→→ �eld, algebra
Combinatorics→→→ linear algebra
In the rest of the talk, k will be an in�nite �eld
Field extensions
Theorem X.D.Hou, K.H.Leung, Q.Xiang (2002)-Kainrath(2005). Let K be a �eld extension of k . Assume every algebraic
element in K is separable over k . Let A and B be two nonempty
�nite subsets of K ∗. Then
dimk(AB) > dimk(A) + dimk(B)− dimk(H) (1)
where H := {h ∈ K | h〈AB〉k−vs ⊆ 〈AB〉k−vs} and dimk(C ) standsfor dimk(〈C 〉k−vs) for every �nite subset C of K .
Main ideas of the proof
Dyson e-transform, Vandermonde determinant, pigeonhole principle
Field extensions
Theorem X.D.Hou, K.H.Leung, Q.Xiang (2002)-Kainrath(2005). Let K be a �eld extension of k . Assume every algebraic
element in K is separable over k . Let A and B be two nonempty
�nite subsets of K ∗. Then
dimk(AB) > dimk(A) + dimk(B)− dimk(H) (1)
where H := {h ∈ K | h〈AB〉k−vs ⊆ 〈AB〉k−vs} and dimk(C ) standsfor dimk(〈C 〉k−vs) for every �nite subset C of K .
Main ideas of the proof
Dyson e-transform,
Vandermonde determinant, pigeonhole principle
Field extensions
Theorem X.D.Hou, K.H.Leung, Q.Xiang (2002)-Kainrath(2005). Let K be a �eld extension of k . Assume every algebraic
element in K is separable over k . Let A and B be two nonempty
�nite subsets of K ∗. Then
dimk(AB) > dimk(A) + dimk(B)− dimk(H) (1)
where H := {h ∈ K | h〈AB〉k−vs ⊆ 〈AB〉k−vs} and dimk(C ) standsfor dimk(〈C 〉k−vs) for every �nite subset C of K .
Main ideas of the proof
Dyson e-transform, Vandermonde determinant,
pigeonhole principle
Field extensions
Theorem X.D.Hou, K.H.Leung, Q.Xiang (2002)-Kainrath(2005). Let K be a �eld extension of k . Assume every algebraic
element in K is separable over k . Let A and B be two nonempty
�nite subsets of K ∗. Then
dimk(AB) > dimk(A) + dimk(B)− dimk(H) (1)
where H := {h ∈ K | h〈AB〉k−vs ⊆ 〈AB〉k−vs} and dimk(C ) standsfor dimk(〈C 〉k−vs) for every �nite subset C of K .
Main ideas of the proof
Dyson e-transform, Vandermonde determinant, pigeonhole principle
Applications
Theorem Kneser's theorem. Let G be an abelian group and
A,B �nite non empty subsets of G , and
H := {g ∈ G , gAB = AB}. We have
|AB| > |A|+ |B| − |H|.
Main ideas of the proof
Realize G = 〈A,B〉gr as the Galois group of a �eld extension (of
�eld of fractions) k ↪→ K , de�ne a map from G to K and use
Galois correspondence theorem.
Applications
Theorem Kneser's theorem. Let G be an abelian group and
A,B �nite non empty subsets of G , and
H := {g ∈ G , gAB = AB}. We have
|AB| > |A|+ |B| − |H|.
Main ideas of the proof
Realize G = 〈A,B〉gr as the Galois group of a �eld extension (of
�eld of fractions) k ↪→ K , de�ne a map from G to K and use
Galois correspondence theorem.
The algebra case
Question : Can we replace K by an associative algebra A in Hou,
Leung and Xiang's or Kainrath's theorem ?
Answer : Of course, not ! Because of zero divisors : we can have
AB = 0 for a big subspaces A and B of A.
→→→Hypothesis on A and B : 〈A〉k−vs and 〈B〉k−vs contain an
invertible element.
→→→ unformal Hypothesis on A : "it is easy to construct invertibles
elements in A".
More precisely, we assume that A is of either one of the following
type �nite dimensional algebras, Banach algebras or �nite products
of �eld extensions of k .
In particular, for all a ∈ A there exists λ ∈ k such that a − λ1 is
invertible in A.
The algebra case
Question : Can we replace K by an associative algebra A in Hou,
Leung and Xiang's or Kainrath's theorem ?
Answer : Of course, not ! Because of zero divisors : we can have
AB = 0 for a big subspaces A and B of A.
→→→Hypothesis on A and B : 〈A〉k−vs and 〈B〉k−vs contain an
invertible element.
→→→ unformal Hypothesis on A : "it is easy to construct invertibles
elements in A".
More precisely, we assume that A is of either one of the following
type �nite dimensional algebras, Banach algebras or �nite products
of �eld extensions of k .
In particular, for all a ∈ A there exists λ ∈ k such that a − λ1 is
invertible in A.
The algebra case
Question : Can we replace K by an associative algebra A in Hou,
Leung and Xiang's or Kainrath's theorem ?
Answer : Of course, not ! Because of zero divisors : we can have
AB = 0 for a big subspaces A and B of A.
→→→Hypothesis on A and B : 〈A〉k−vs and 〈B〉k−vs contain an
invertible element.
→→→ unformal Hypothesis on A : "it is easy to construct invertibles
elements in A".
More precisely, we assume that A is of either one of the following
type �nite dimensional algebras, Banach algebras or �nite products
of �eld extensions of k .
In particular, for all a ∈ A there exists λ ∈ k such that a − λ1 is
invertible in A.
The algebra case
Question : Can we replace K by an associative algebra A in Hou,
Leung and Xiang's or Kainrath's theorem ?
Answer : Of course, not ! Because of zero divisors : we can have
AB = 0 for a big subspaces A and B of A.
→→→Hypothesis on A and B : 〈A〉k−vs and 〈B〉k−vs contain an
invertible element.
→→→ unformal Hypothesis on A : "it is easy to construct invertibles
elements in A".
More precisely, we assume that A is of either one of the following
type �nite dimensional algebras, Banach algebras or �nite products
of �eld extensions of k .
In particular, for all a ∈ A there exists λ ∈ k such that a − λ1 is
invertible in A.
The algebra case
Question : Can we replace K by an associative algebra A in Hou,
Leung and Xiang's or Kainrath's theorem ?
Answer : Of course, not ! Because of zero divisors : we can have
AB = 0 for a big subspaces A and B of A.
→→→Hypothesis on A and B : 〈A〉k−vs and 〈B〉k−vs contain an
invertible element.
→→→ unformal Hypothesis on A : "it is easy to construct invertibles
elements in A".
More precisely, we assume that A is of either one of the following
type �nite dimensional algebras, Banach algebras or �nite products
of �eld extensions of k .
In particular, for all a ∈ A there exists λ ∈ k such that a − λ1 is
invertible in A.
Diderrich's theorem for algebras
Theorem B., Lecouvey (2015). Assume A is isomorphic to
a subalgebra of an algebra of the three preceeding types.
Assume A and B to be two �nite nonempty subsets of A such that
〈A〉k−vs. ∩ U(A) 6= ∅ and 〈B〉k−vs. ∩ U(A) 6= ∅.
Assume that A is commutative and A(A) admits a �nite number of
�nite-dimensional subalgebras.
Let H := {x ∈ A, x〈AB〉k−vs ⊂ 〈AB〉k−vs}.We have
dimk(AB) > dimk(A) + dimk(B)− dim(H)
Sketch of proof of the theorem
Lemma Assume A is of one of the three preceeding types.
Let A and B be two �nite subsets of A such that A is
commutative, 〈A〉k−vs. ∩ U(A) 6= ∅ and 〈B〉k−vs. ∩ U(A) 6= ∅.
Then, for each a ∈ 〈A〉k−vs ∩ U(A), there exists a (commutative)
�nite-dimensional subalgebra Aa of A such that Aa ⊆ A(A) and a
vector space Va contained in 〈AB〉k−vs such that Va ∩ U(A) 6= ∅,
AaVa = Va, 〈aB〉k−vs ⊆ Va and
dimk(Va) + dimk(Aa) > dimk(A) + dimk(B).
Proof of Lemma e-Dyson transform, recursion on dimension of
〈A〉k−vs ; the type of A ensures us that at each step there are
enough invertible elements in 〈A〉k−vs .
Sketch of proof of the theorem
Lemma Assume A is of one of the three preceeding types.
Let A and B be two �nite subsets of A such that A is
commutative, 〈A〉k−vs. ∩ U(A) 6= ∅ and 〈B〉k−vs. ∩ U(A) 6= ∅.
Then, for each a ∈ 〈A〉k−vs ∩ U(A), there exists a (commutative)
�nite-dimensional subalgebra Aa of A such that Aa ⊆ A(A) and a
vector space Va contained in 〈AB〉k−vs such that Va ∩ U(A) 6= ∅,
AaVa = Va, 〈aB〉k−vs ⊆ Va and
dimk(Va) + dimk(Aa) > dimk(A) + dimk(B).
Proof of Lemma e-Dyson transform, recursion on dimension of
〈A〉k−vs ; the type of A ensures us that at each step there are
enough invertible elements in 〈A〉k−vs .
Sketch of proof of the theorem
Let (x1, . . . , xn) be a basis of 〈A〉k−vs with x1 invertible.
For any α ∈ k , set xα = x1 + αx2 + · · ·+ αn−1xn.
For each α such that xα is invertible (there exists an in�nity of such
α), there exists a �nite-dimensional subalgebra Aα and a subspace
Vα such in the preceeding lemma.
Since A(A) has only a �nite number of subalgebras, there exists n
distincts elements of k , α1, . . . , αn such that
B := Aα1 = · · · = Aαn . Moreover (xα1 , . . . , xαn) is a basis of
〈A〉k−vs .
We have B ⊂ H.
A �rst example
Example Let C the Banach algebra of continuous functions from
[ 0 , 1 ] into R. Let V and W be �nite dimensional subspaces
containing a positive function. We have
dimR(VW ) > dimR(V ) + dimR(W )− 1.
Back to Kneser-Diderrich's theorem
for groups
Let G be a group and A,B �nite non empty subsets of G . Assume
A is a commutative subset of G and let H := {g ∈ G , gAB = AB}.
|AB| > |A|+ |B| − |H|.
Going to algebras. We consider A = C[G ]. This is a subalgebra of a
Banach algebra.
The algebra A(A)C-alg.' C[X±1
1, . . . ,Xn
±1]r has only a �nite number
of �nite dimensional subalgebras.
The stabilizer of 〈AB〉C−vs in C[G ] is C[H].
Diderrich's theorem in C[G ] gives Didderich's theorem for G .
Back to Kneser-Diderrich's theorem
for groups
Let G be a group and A,B �nite non empty subsets of G . Assume
A is a commutative subset of G and let H := {g ∈ G , gAB = AB}.
|AB| > |A|+ |B| − |H|.
Going to algebras. We consider A = C[G ]. This is a subalgebra of a
Banach algebra.
The algebra A(A)C-alg.' C[X±1
1, . . . ,Xn
±1]r has only a �nite number
of �nite dimensional subalgebras.
The stabilizer of 〈AB〉C−vs in C[G ] is C[H].
Diderrich's theorem in C[G ] gives Didderich's theorem for G .