Linear groups over associative rings.

A.V. Mikhalev

Faculty of mechanics and mathematics Moscow 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 of the 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 all diagonal matrices.

Definition

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

A.V. Mikhalev Linear groups over associative rings. May 2013 2 / 23 History

In 1980s, I.Z. Golubchik, A.V. Mikhalev and E.I. Zelmanov described isomorphisms of general linear groups GLn(R) over associative rings 1 with 2 for n > 3.

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

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

A.V. Mikhalev Linear groups over associative rings. May 2013 3 / 23 History: main result

Theorem (I.Z. Golubchik and A.V. Mikhalev) Let R and S be associative rings with unit, n > 4, m > 2 and ϕ : GLn(R) −→ GLm(S) be a group isomorphism. Then there exist central idempotents e and f of the rings Mat n(R) and Mat m(S) respectively, a ring isomorphism θ1 : eMat n(R) → f Mat m(S), a ring anti-isomorphism

θ2 : (1 − e)Mat n(R) → (1 − f )Mat m(S),

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

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

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

A.V. Mikhalev Linear groups over associative rings. May 2013 4 / 23 Basic definitions of the graded rings theory

Definition A ring R is called G-graded if

R = R , M g g∈G

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

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

A.V. Mikhalev Linear groups over associative rings. May 2013 5 / 23 Basic definitions of the graded modules theory

Definition

A right R-module M is called G-graded if M = Mg , where gL∈G {Mg | g ∈ G} is a system of additive subgroups in M such that MhRg ⊆ Mhg for all h, g ∈ G.

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

A.V. Mikhalev Linear groups over associative rings. May 2013 6 / 23 Basic definitions of the graded modules theory

Definition Let END (M) := HOM (M, M) . R M R g g∈G Then this graded ring is called the graded endomorphism ring of the graded R-module M.

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

A.V. Mikhalev Linear groups over associative rings. May 2013 7 / 23 Description of graded endomorphism rings

Let R = Rg be an associative graded ring with 1, M be a finitely gL∈G generated gr-free right R-module with a basis consisting of homogeneous

elements v1, v2,..., vn where vi ∈ Mgi (i = 1,..., n). Then the graded endomorphism ring END R (M) is isomorphic to the graded matrix ring

Mat (R)(g1,..., g )= Mat (R) (g1,..., g ), n n M n h n h∈G

where

R −1 R −1 . . . R −1  g1 hg1 g1 hg2 g1 hgn  R −1 R −1 . . . R −1 g2 hg1 g2 hg2 g2 hgn Mat n(R)h(g1,..., gn)=   .  . . .. .   . . . .    R −1 R −1 . . . R −1  gn hg1 gn hg2 gn hgn 

A.V. Mikhalev Linear groups over associative rings. May 2013 8 / 23 An isomorphism respecting grading

We introduce the following notion. Definition

Let R = Rg and S = Sg be associative graded rings with 1, gL∈G gL∈G 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

and A − E ∈ Mat n(R)g =⇒ ϕ(A) − E ∈ Mat m(S)g .

A.V. Mikhalev Linear groups over associative rings. May 2013 9 / 23 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 = Rg and S = Sg are gL∈G gL∈G associative graded rings with 1, Mat n(R), Mat m(S) are graded matrix rings, 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) and Mat m(S) respectively, e ∈ Mat n(R)0, f ∈ Mat m(S)0, a ring isomorphism θ1 : eMat n(R) −→ f Mat m(S) and a ring anti-isomorphism θ2 : (1 − e)Mat n(R) −→ (1 − f )Mat m(S), both of them preserve grading, such that

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

Remark. Also according to Baer–Kaplansky graded Theorem proved by A.V. Mikhalev and I.N. Balaba all isomorphisms and anti-isomorphisms of graded matrix rings are completely described. A.V. Mikhalev Linear groups over associative rings. May 2013 10 / 23 Stable linear groups. Basic definitions.

Denote by Mat ∞(R) the set of all matrices with countable number of lines and rows but with finite number of nonzero elements outside of the main diagonal and such that there exists a number n with the property that for every 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 the following 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)= GL (R). [ n n>1

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

A.V. Mikhalev Linear groups over associative rings. May 2013 11 / 23 The stable linear groups. Basic definitions.

As above, we can include into Mat ∞(R) the subgroups of elementary matrices E n(R). Definition Let us set E (R)= E (R) [ n n>1

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

A.V. Mikhalev Linear groups over associative rings. May 2013 12 / 23 Isomorphisms of the stable linear groups over rings

Li Fuan, 1994: Automorphisms of stable linear groups over arbitrary commutative rings We describe the action of a stable linear groups isomorphism on the stable elementary subgroup. Theorem (A.S. Atkarskaya, 2013) 1 Let R and S be associative rings with 2 , ϕ : GL(R) → GL(S) be a group isomorphism. Then there exist central idempotents h and e of the rings Mat (R) and Mat (S) respectively, a ring isomorphism θ1 : hMat (R) → eMat (S) and a ring antiisomorphism θ2 : (1 − h)Mat (R) → (1 − e)Mat (S) such that

−1 ϕ(A)= θ1(hA)+ θ2((1 − h)A )

for all A ∈ E (R).

A.V. Mikhalev Linear groups over associative rings. May 2013 13 / 23 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-known hypothesis:

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

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

A.V. Mikhalev Linear groups over associative rings. May 2013 14 / 23 An analogue of Gerasimov theorem for Unitary linear groups

We consider Unitary linear groups U2n(R, j, Q) over rings R with involutions j with the form Q of maximal rang. Its elementary subgroup UE2n(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) ∗F2 H,

where H is a nontrivial subgroup of U2n(R, j, Q), n > 2 is a given natural number. Now we generalize this theorem for an arbitrary field Λ.

A.V. Mikhalev Linear groups over associative rings. May 2013 15 / 23 Elementary equivalence, Maltsev Theorem

Definition (Elementary equivalence) Two models U and U′ of the same first order language are called elementary equivalent (notation: U ≡ U′), if for every first order sentence ϕ 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 if m = n, fields K and K ′ elementary equivalent.

A.V. Mikhalev Linear groups over associative rings. May 2013 16 / 23 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 an ultrafilter F such that ′ U =∼ U . Y Y F F

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

A.V. Mikhalev Linear groups over associative rings. May 2013 17 / 23 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 ′ are elementary 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 the ′ ′ op matrix rings Mn(K) and Mm(K ), or Mn(K) Mm(K ) are elementary equivalent.

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

A.V. Mikhalev Linear groups over associative rings. May 2013 18 / 23 Elementary equivalence of unitary linear groups

E.I. Bunina, 1998. Extension of Maltsev Theorem to unitary linear groups over 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′) elementary equivalent if and only if m = n, and (K, j) and (K ′, j′) are elementary equivalent 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 with involutions τ and τ ′, induced by involutions j and j′.

A.V. Mikhalev Linear groups over associative rings. May 2013 19 / 23 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 and weight 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), ...

A.V. Mikhalev Linear groups over associative rings. May 2013 20 / 23 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 and Φ′, respectively; π, π′ are finitely dimensional complex representations of algebras 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 ′.

A.V. Mikhalev Linear groups over associative rings. May 2013 21 / 23 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 root systems Φ and Φ′, respectively; π, π′ are finitely dimensional complex representations of algebras L and L′, respectively, with weight lattices Λ and Λ′; R and R′ are local commutative rings with 1. Suppose that every system Φ, Φ′ has at least one irreducible component of rank > 1; R, R′ contain 1/2. Then ′ ′ Gπ(Φ, R) ≡ Gπ′ (Φ , R ) if and only if

Φ=Φ′, Λ=Λ′, R ≡ R′.

A.V. Mikhalev Linear groups over associative rings. May 2013 22 / 23 Other results of this type.

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

(E.I. Bunina, P.P. Semenov, 2008) Elementary equivalence of semigroups of invertible nonnegative matrices over partially ordered commutative rings.

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

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