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 GL n ( R ) generated by the matrices E + re ij , i � = j . Definition The group D n ( R ) is the subgroup of the group GL n ( R ) generated by all diagonal matrices. Definition The group GE n ( R ) is the subgroup of the group GL n ( R ) generated by the subgroups E n ( R ) and D n ( 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 GL n ( R ) over associative rings with 1 2 for n � 3 . In 1997, I.Z. Golubchik and A.V. Mikhalev described isomorphisms of the group GL n ( 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 : GL n ( R ) − → GL m ( 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 θ 1 : e Mat n ( R ) → f Mat m ( S ) , a ring anti-isomorphism θ 2 : (1 − e ) Mat n ( R ) → (1 − f ) Mat m ( S ) , and a group homomorphism χ : GE n ( R ) → Z (GL m ( S )) such that ϕ ( A ) = χ ( A )( θ 1 ( eA ) + θ 2 ((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 = R g , g ∈ G where { R g | g ∈ G } is a system of additive subgroups of the ring R and R g R h ⊆ R gh for all g , h ∈ G . If R s R h = R sh 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 ( R g ) ∼ = S g 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 = � M g , where g ∈ G { M g | g ∈ G } is a system of additive subgroups in M such that M h R g ⊆ M hg 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 ( M h ) ⊆ N gh 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 ) := 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 = � R g 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 v 1 , v 2 , . . . , v n where v i ∈ M g i ( i = 1 , . . . , n ) . Then the graded endomorphism ring END R ( M ) is isomorphic to the graded matrix ring � Mat n ( R )( g 1 , . . . , g n ) = Mat n ( R ) h ( g 1 , . . . , g n ) , h ∈ G where R g − 1 R g − 1 . . . R g − 1 hg 1 hg 2 hg n 1 1 1 . . . R g − 1 R g − 1 R g − 1 hg 1 hg 2 hg n Mat n ( R ) h ( g 1 , . . . , g n ) = 2 2 2 . . . . ... . . . . . . . . . R g − 1 R g − 1 R g − 1 hg 1 hg 2 hg n n n n A.V. Mikhalev Linear groups over associative rings. May 2013 8 / 23
An isomorphism respecting grading We introduce the following notion. Definition Let R = � R g and S = � S g be associative graded rings with 1 , g ∈ G g ∈ G Mat n ( R ) , Mat n ( S ) be graded matrix rings. A group isomorphism ϕ : GL n ( R ) − → GL m ( S ) is called an isomorphism respecting grading , if ϕ (GL n ( R ) ∩ Mat n ( R ) e ) ⊆ GL m ( 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 = � R g and S = � S g are g ∈ G g ∈ G associative graded rings with 1 , Mat n ( R ) , Mat m ( S ) are graded matrix rings, n � 4 , m � 4 , and ϕ : GL n ( R ) − → GL m ( 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 θ 1 : e Mat n ( R ) − → f Mat m ( S ) and a ring anti-isomorphism θ 2 : (1 − e ) Mat n ( R ) − → (1 − f ) Mat m ( S ) , both of them preserve grading, such that ϕ ( A ) = θ 1 ( eA ) + θ 2 ((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 r ii = a , a ∈ R . Definition Let A ∈ GL n ( 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 n ( 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 1 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 θ 1 : h Mat ( R ) → e Mat ( S ) and a ring antiisomorphism θ 2 : (1 − h ) Mat ( R ) → (1 − e ) Mat ( S ) such that ϕ ( A ) = θ 1 ( hA ) + θ 2 ((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 GL n ( R ) = GE n ( 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 E n ( R ) is always normal in GL n ( R ) . 2 Any automorphism of GL n ( 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 U 2 n ( R , j , Q ) over rings R with involutions j with the form Q of maximal rang. Its elementary subgroup UE 2 n ( R , j , Q ) is generated by unitary transvections . Theorem (M.V. Tsvetkov, 2013) There exists an algebra R over the field F 2 such that U 2 n ( R , j , Q ) = UE 2 n ( R , j , Q ) ∗ F ∗ 2 H , where H is a nontrivial subgroup of U 2 n ( 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
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