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
with 12 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
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. 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=
M
Rg ,
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 =
L
Mg , where
g ∈G
{Mg | g ∈ G } is a system of additive subgroups in M such that
Mh Rg ⊆ 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 R (M) :=
M
HOM R (M, M)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 =
L
Rg be an associative graded ring with 1, M be a finitely
g ∈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
M
Mat n (R)(g1 , . . . , gn ) =
Mat n (R)h (g1 , . . . , gn ),
h∈G
where

Rg −1 hg
 1 1
Rg −1 hg1
2
Mat n (R)h (g1 , . . . , gn ) = 
 ..
 .
Rgn−1 hg1
A.V. Mikhalev
Rg −1 hg2
1
Rg −1 hg2
2
..
.
Rgn−1 hg2
Linear groups over associative rings.
...
...
..
.
...

Rg −1 hgn
1

Rg −1 hgn 
2

..  .
. 
Rgn−1 hgn
May 2013
8 / 23
An isomorphism respecting grading
We introduce the following notion.
Definition
Let R =
L
Rg and S =
L
Sg be associative graded rings with 1,
g ∈G
g ∈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 =
L
g ∈G
Rg and S =
L
Sg are
g ∈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
ϕ(A) = θ1 (eA) + θ2 ((1 − e)A−1 ) 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) =
GLn (R).
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 n (R)
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)
Let R and S be associative rings with 12 , 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
ϕ(A) = θ1 (hA) + θ2 ((1 − h)A−1 )
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) ∗F∗2 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
Y
Y
U ′.
U∼
=
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
matrix 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 of
the modules R n and S m 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