Linear groups over associative rings. · Linear groups over associative rings. A.V. Mikhalev Faculty of mechanics and mathematics Moscow State University May 2013 A.V. Mikhalev Linear

Linear groups over associative rings.

A.V. Mikhalev

Faculty of mechanics and mathematicsMoscow State University

May 2013

A.V. Mikhalev Linear groups over associative rings. May 2013 1 / 23

Main definitions

Definition

For arbitrary associative ring R with 1 the group E n(R) is the subgroup ofthe group GLn(R) generated by the matrices E + reij , i 6= j .

Definition

The group Dn(R) is the subgroup of the group GLn(R) generated by alldiagonal matrices.

Definition

The group GEn(R) is the subgroup of the group GLn(R) generated by thesubgroups E n(R) and Dn(R).


History

In 1980s, I.Z. Golubchik, A.V. Mikhalev and E.I. Zelmanov describedisomorphisms of general linear groups GLn(R) over associative ringswith 1

2for n > 3.

In 1997, I.Z. Golubchik and A.V. Mikhalev described isomorphisms ofthe group GLn(R) over arbitrary associative rings, n > 4.

2000–2012: extensions of these theorems for various linear groups overdifferent types of rings.


History: main result

Theorem (I.Z. Golubchik and A.V. Mikhalev)

Let R and S be associative rings with unit, n > 4, m > 2 andϕ : GLn(R) −→ GLm(S) be a group isomorphism. Then there exist centralidempotents e and f of the rings Mat n(R) and Matm(S) respectively, aring isomorphism θ1 : eMat n(R) → fMatm(S), a ring anti-isomorphism

θ2 : (1− e)Mat n(R) → (1− f )Matm(S),

and a group homomorphism χ : GE n(R) → Z (GLm(S)) such that

ϕ(A) = χ(A)(θ1(eA) + θ2((1− e)A−1)) for all A ∈ GE n(R).

Remark. According to Baer–Kaplansky Theorem proved by A.V. Mikhalevfor modules close to free modules all isomorphisms and anti-isomorphismsof matrix rings are completely described.


Basic definitions of the graded rings theory

Definition

A ring R is called G -graded if

R =⊕

g∈G

Rg ,

where {Rg | g ∈ G} is a system of additive subgroups of the ring R andRgRh ⊆ Rgh for all g , h ∈ G . If RsRh = Rsh for all s, h ∈ G , then the ringis called strongly graded.

Definition

Two G -graded rings R and S are called isomorphic if there exists a ringisomorphism f : R → S such that f (Rg ) ∼= Sg for all g ∈ G .


Basic definitions of the graded modules theory

Definition

A right R-module M is called G -graded if M =⊕

g∈G

Mg , where

{Mg | g ∈ G} is a system of additive subgroups in M such thatMhRg ⊆ Mhg for all h, g ∈ G .

Definition

An R-linear map f : M → N of right G -graded R-modules is called agraded morphism of degree g , if f (Mh) ⊆ Ngh for all h ∈ G . The set ofgraded morphisms of degree g is the subgroup HOM R(M,N)g of thegroup Hom R(M,N).


Basic definitions of the graded modules theory

Definition

LetEND R(M) :=

⊕

g∈G

HOM R(M,M)g .

Then this graded ring is called the graded endomorphism ring of the gradedR-module M.

Definition

A graded right R-module M is called gr-free, if there exists a basis thatconsists of homogeneous elements.


Description of graded endomorphism rings

Let R =⊕

g∈G

Rg be an associative graded ring with 1, M be a finitely

generated gr-free right R-module with a basis consisting of homogeneouselements v1, v2, . . . , vn where vi ∈ Mgi (i = 1, . . . , n). Then the gradedendomorphism ring END R(M) is isomorphic to the graded matrix ring

Mat n(R)(g1, . . . , gn) =⊕

h∈G

Mat n(R)h(g1, . . . , gn),

where

Mat n(R)h(g1, . . . , gn) =

Rg−1

1hg1

Rg−1

1hg2

. . . Rg−1

1hgn

Rg−1

2hg1

Rg−1

2hg2

. . . Rg−1

2hgn

......

. . ....

Rg−1n hg1

Rg−1n hg2

. . . Rg−1n hgn

.


An isomorphism respecting grading

We introduce the following notion.

Definition

Let R =⊕

g∈G

Rg and S =⊕

g∈G

Sg be associative graded rings with 1,

Mat n(R), Mat n(S) be graded matrix rings. A group isomorphism

ϕ : GLn(R) −→ GLm(S)

is called an isomorphism respecting grading, if

ϕ(GLn(R) ∩Mat n(R)e) ⊆ GLm(S)e

andA− E ∈ Mat n(R)g =⇒ ϕ(A)− E ∈ Matm(S)g .


Isomorphisms of linear groups over associative graded rings

Theorem (A.S. Atkarskaya, E.I. Bunina, A.V. Mikhalev, 2009)

Suppose that G is a commutative group, R =⊕

g∈G

Rg and S =⊕

g∈G

Sg are

associative graded rings with 1, Mat n(R), Matm(S) are graded matrixrings, n > 4, m > 4, and ϕ : GLn(R) −→ GLm(S) is a group isomorphism,respecting grading. Suppose that ϕ−1 also respects grading.Then there exist central idempotents e and f of the rings Mat n(R) andMatm(S) respectively, e ∈ Mat n(R)0, f ∈ Matm(S)0, a ring isomorphismθ1 : eMat n(R) −→ fMatm(S) and a ring anti-isomorphismθ2 : (1− e)Mat n(R) −→ (1− f )Matm(S), both of them preserve grading,such that

ϕ(A) = θ1(eA) + θ2((1− e)A−1) for all A ∈ E n(R).

Remark. Also according to Baer–Kaplansky graded Theorem proved byA.V. Mikhalev and I.N. Balaba all isomorphisms and anti-isomorphisms ofgraded matrix rings are completely described.


Stable linear groups. Basic definitions.

Denote by Mat∞(R) the set of all matrices with countable number of linesand rows but with finite number of nonzero elements outside of the maindiagonal and such that there exists a number n with the property that forevery i > n the elements of our matrix rii = a, a ∈ R .

Definition

Let A ∈ GLn(R). We identify A with an element from Mat∞(R) by thefollowing rule: A is placed into the left upper corner, and from the position(n, n) we place 1 on the diagonal, and 0 in all other positions.Let us set

GL(R) =⋃

n>1

GLn(R).

It is a subgroup of the group of all invertible elements of Mat∞(R). It iscalled the stable linear group.


The stable linear groups. Basic definitions.

As above, we can include into Mat∞(R) the subgroups of elementarymatrices E n(R).

Definition

Let us setE (R) =

⋃

n>1

E n(R)

(E n(R) ⊆ Mat∞(R)). It is a subgroup of the group of all invertibleelements of Mat∞(R). We call it the stable elementary group.


Isomorphisms of the stable linear groups over rings

Li Fuan, 1994: Automorphisms of stable linear groups over arbitrarycommutative rings

We describe the action of a stable linear groups isomorphism on the stableelementary subgroup.

Theorem (A.S. Atkarskaya, 2013)

Let R and S be associative rings with 1

2, ϕ : GL(R) → GL(S) be a group

isomorphism. Then there exist central idempotents h and e of the ringsMat (R) and Mat (S) respectively, a ring isomorphismθ1 : hMat (R) → eMat (S) and a ring antiisomorphismθ2 : (1− h)Mat (R) → (1− e)Mat (S) such that

ϕ(A) = θ1(hA) + θ2((1− h)A−1)

for all A ∈ E (R).


Rings where the elementary subgroup is a free multiplier in

the whole linear group

Theorem (V.N. Gerasimov, 1987)

There exists an algebra R over a given field Λ such that

GLn(R) = GEn(R) ∗Λ∗ H,

where H is a subgroup not equal to Λ∗, n > 2 is a given natural number.

Every such algebra is a counter example to the following two well-knownhypothesis:

1 The subgroup En(R) is always normal in GLn(R).

2 Any automorphism of GLn(R) (n > 3) is standard.


An analogue of Gerasimov theorem for Unitary linear groups

We consider Unitary linear groups U2n(R , j ,Q) over rings R withinvolutions j with the form Q of maximal rang. Its elementary subgroupUE2n(R , j ,Q) is generated by unitary transvections.

Theorem (M.V. Tsvetkov, 2013)

There exists an algebra R over the field F2 such that

U2n(R , j ,Q) = UE2n(R , j ,Q) ∗F∗

2H,

where H is a nontrivial subgroup of U2n(R , j ,Q), n > 2 is a given naturalnumber.

Now we generalize this theorem for an arbitrary field Λ.


Elementary equivalence, Maltsev Theorem

Definition (Elementary equivalence)

Two models U and U ′ of the same first order language are calledelementary equivalent (notation: U ≡ U ′), if for every first order sentenceϕ of this language ϕ holds in U if and only if it holds in U ′.

If U ∼= U ′, then U ≡ U ′.

If U ≡ U ′ and U is finite, then U ∼= U ′.

C ≡ Q, but C 6∼= Q.

Theorem (A.I. Maltsev, 1961.)

Two groups Gn(K ) and Gm(K′) (where G = GL,SL,PGL,PSL, n,m > 3,

K ,K ′ are fields of characteristics 0) elementary equivalent if and only ifm = n, fields K and K ′ elementary equivalent.


Keisler–Shelah and Beidar–Mikhalev Theorems

Theorem (Keisler–Shelah Isomorphism Theorem, 1971–1974)

Two models U and U ′ elementary equivalent if and only if there exists anultrafilter F such that

∏

F

U ∼=∏

F

U ′.

=⇒ K.I. Beidar and A.V. Mikhalev, 1992. Generalizations of MaltsevTheorem for the cases when K and K ′ are skewfields, associative rings;similar theorems for different algebraic structures (endomorphism rings,lattices of submodules):


Beidar–Mikhalev Theorem, 1992

Theorem (Linear groups over skewfields)

Linear groups GLn(K ) and GLm(K′) (n,m > 3, K ,K ′ are skewfields) are

elementary equivalent if and only if m = n and either K and K ′ areelementary equivalent, or K and K ′op elementary equivalent.

Theorem (Linear groups over prime rings)

Groups GLn(K ) and GLm(K′) (K ,K ′ are prime associative rings with 1 or

1/2, n,m > 4 or n,m > 3, elementary equivalent if and only if either thematrix rings Mn(K ) and Mm(K

′), or Mn(K ) Mm(K′)op are elementary

equivalent.

Theorem (Lattices of submodules)

If R and S are associative rings with 1, m, n > 3, lattices of submodules ofthe modules Rn and Sm are elementary equivalent, then the matrix ringsMn(R) and Mm(S) are elementary equivalent.


Elementary equivalence of unitary linear groups

E.I. Bunina, 1998. Extension of Maltsev Theorem to unitary linear groupsover skewfields and associative rings with involutions:

Theorem (Unitary groups over skewfields)

Unitary linear groups U2n(K , j ,Q2n) and U2m(K′, j ′,Q2m) (n,m > 3, K ,K ′

are skewfields of characteristics 6= 2, with involutions j , j ′) elementaryequivalent if and only if m = n, and (K , j) and (K ′, j ′) are elementaryequivalent as skewfields with involutions.

Theorem (Unitary groups over rings)

Unitary linear groups U2n(K , j ,Q2n) and U2m(K′, j ′,Q2m), where K ,K ′ are

associative (commutative) rings with 1/6, with involutions j , j ′, n,m > 3(n,m > 2), are elementary equivalent if and only if matrix rings(M2n(K ), τ) and (M2m(K

′), τ ′) are elementary equivalent as rings withinvolutions τ and τ ′, induced by involutions j and j ′.


Chevalley groups.

Definition (Chevalley groups)

Every Chevalley group Gπ(R ,Φ) is constructed by:— a semisimple complex Lie algebra L with a root system Φ;— a linear representation π : L → GLN(C);— a commutative ring R Гҫ 1.A group Gπ(R ,Φ) is defined by a commutative ring R , root system Φ andweight lattice Λπ of the representation π.

Example

Al — SLl+1(R), PGLl+1(R), . . . ;

Bl — Spin2l+1(R), SO2n+1(R);

Cl — Sp2l(R), PSp2l(R);

Dl — Spin2l(R), SO2l (R), PSO2l (R), . . .


Elementary equivalence of Chevalley groups over fields.

Theorem (E.I. Bunina, 2004)

Suppose that L and L′ are complex Lie algebras with root systems Φ andΦ′, respectively; π, π′ are finitely dimensional complex representations ofalgebras L and L′, respectively, with weight lattices Λ and Λ′; K and K ′

are fields of characteristics 6= 2.Then

Gπ(Φ,K ) ≡ Gπ′(Φ′,K ′)

if and only if

Φ = Φ′,

Λ = Λ′,

K ≡ K ′.


Elementary equivalence of Chevalley groups over local rings.

Theorem (E.I. Bunina, 2006–2009)

Suppose that L and L′ are complex semisimple Lie algebras with rootsystems Φ and Φ′, respectively; π, π′ are finitely dimensional complexrepresentations of algebras L and L′, respectively, with weight lattices Λand Λ′; R and R ′ are local commutative rings with 1.Suppose that every system Φ, Φ′ has at least one irreducible component ofrank > 1; R , R ′ contain 1/2.Then

Gπ(Φ,R) ≡ Gπ′(Φ′,R ′)

if and only if

Φ = Φ′,

Λ = Λ′,

R ≡ R ′.


Other results of this type.

(E.I. Bunina, A.V. Mikhalev, 2004) Elementary equivalence ofsemigroups of invertible nonnegative matrices over linearly orderedassociative rings.

(E.I. Bunina, P.P. Semenov, 2008) Elementary equivalence ofsemigroups of invertible nonnegative matrices over partially orderedcommutative rings.

(E.I. Bunina, A.S. Dobrokhotova–Maykova, 2009) Elementaryequivalence of incidence rings over semi-perfect rings.


Documents

Linear groups over associative rings. · Linear groups over associative rings. A.V. Mikhalev Faculty of mechanics and mathematics Moscow State University May 2013 A.V. Mikhalev Linear