Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 242736, 12 pages
doi:10.1155/2012/242736
Research Article
Decomposition of Automorphisms of
Certain Solvable Subalgebra of Symplectic Lie
Algebra over Commutative Rings
Xing Tao Wang and Lei Zhang
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Xing Tao Wang, xingtao@hope.hit.edu.cn
Received 26 March 2012; Accepted 26 April 2012
Academic Editor: Helge Holden
Copyright q 2012 X. T. Wang and L. Zhang. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let Cl1 R be the 2l 1 × 2l 1 matrix symplectic Lie algebra over a commutative ring R with
m m
C
2 invertible. Then tl1 R { 01 −m2T | m1 is an l 1 upper triangular matrix, mT2 m2 , over
1
R} is the solvable subalgebra of Cl1 R. In this paper, we give an explicit description of the autoC
morphism group of tl1 R.
1. Introduction
Classical Lie algebras occupy an important place in matrix algebras. Let R be a commutative
ring R with the identity 1 and R∗ the group of invertible elements in R. Let Mn R be the
R-algebra of n by n matrices over R that has a structure of a Lie algebra over R with bracket
operation x, y xy − yx for any x, y ∈ Mn R. The symplectic Lie algebra
Cl1 R X|X
0 Il
−Il 0
0 Il
X T 0, X ∈ M2l2 R
−Il 0
1.1
is one of classical Lie algebras, where T denotes the matrix transpose. It is easy to show that
the following subalgebra of Cl1 R such that
C
tl1 R
is solvable.
m1 m2
0 −mT1
| m1 is an l 1 upper triangular
matrix, mT2
m2
1.2
2
International Journal of Mathematics and Mathematical Sciences
n
Let e be the identity matrix in Mn R and let eij denotes the matrix in Mn R all of
whose entries are 0, except the i, jth entry which is 1. Let
n
n
αi,ik ei,ik − elik1,li1 ,
n
n
γi,ik ei,lik1 eik,li1 ,
n
γii ei,li1 ,
i 1, . . . , l − k 1, k 0, 1, . . . , l,
i 1, . . . , l − k 1, k 1, . . . , l,
1.3
i 1, . . . , l 1,
C
where n 2l 1, l ≥ 1. For discussion latter, we rewrite tl1 R as
C
tl1 R l l−k1
l l−k1
Rαi,ik Rγi,ik .
k0 i1
1.4
k0 i1
Automorphisms of associative algebras have been explored in many articles 1–8.
Encouraged by Doković 9 and Cao’s 10 papers which described the automorphism groups
of Lie algebra consisting of all upper triangular n × n matrices of trace 0 over a connected
commutative ring and a commutative ring with n invertible, respectively, in this paper we
C
use similar techniques to those in 11 to prove that any automorphism ψ of tl1 R can be
uniquely expressed as ψ θλD , where θ and λD are inner and diagonal automorphisms,
respectively, for l ≥ 1 and R is a commutative ring with 2 invertible. We also give an explicit
description of the remaining case l 0.
C
Theorem 1.1. For any automorphism ψ of tl1 R l ≥ 1 there are unique inner and diagonal autoC
morphisms, θ and λD , respectively, of tl1 R such that ψ θλD .
Theorem 1.2. Let I and D be the inner and diagonal automorphism groups, respectively. Then
C
C
C
Auttl1 R I D, where Auttl1 R denotes the automorphism group of tl1 R.
2. Preliminaries
Let
Pn {αi,ik | i 1, . . . , l − k 1, k 0, 1, . . . , l},
Wn γi,ik | i 1, . . . , l − k 1, k 0, 1, . . . , l .
2.1
C
Then the set Pn ∪ Wn is a basis of tl1 R.
Lemma 2.1. Let Hn be the set generated by the set {αjj , αi,i1 , γl1,l1 | 1 ≤ j ≤ l 1, 1 ≤ i ≤ l}, where
C
n 2l 1. Then Hn tl1 R.
International Journal of Mathematics and Mathematical Sciences
3
C
Proof. We only need to show that tl1 R ⊆ Hn . It is obvious that αi,ik ∈ Hn , when k 0, 1.
When k 2, we have
n
n
αi,i2 ei,i2 − eli3,li1
n
n
n
n
ei,i1 − eli2,li1 , ei1,i2 − eli3,li2
2.2
αi,i1 , αi1,i2 ∈ Hn .
Assume that αi,ik−1 ∈ Hn , then αi,ik αi,ik−1 , αik−1,ik ∈ Hn , that is, Pn ⊆ Hn . Since γl1,l1 ∈
Hn , for any γl−k1,l−k2 ∈ Wn , when k 1,
n
n
γl,l1 el,2l2 el1,2l1
n
n
n
el,l1 − e2l2,2l1 , el1,2l2
2.3
αl,l1 , γl1,l1 ∈ Hn .
Assume that when k m − 1, γl−m2,l−m3 ∈ Hn , then when k m, γl−m1,l−m2 αl−m1,l−m3 ,
γl−m2,l−m3 ∈ Hn , that is, γi,i1 ∈ Hn , i 1, . . . , l. For any γi,ik ∈ Hn k ≥ 1, when k 1,
γi,i1 ∈ Hn . When k ≥ 2, γi,ik αi,ik−1 , γik−1,ik ∈ Hn . Since 2γii αi,i1 , γi,i1 ∈ Hn , and 2 is
C
invertible, we have γii ∈ Hn for 1 ≤ i ≤ l. Thus Wn ⊆ Hn . Because Pn ∪ Wn is a basis of tl1 R,
C
we obtain tl1 R ⊆ Hn .
C
C
C
C
Now, denote tl1 R by n0 . Let n1
C
C
C
n0 , n0 , n2
C
C
C
n1 , n1 , nj
C
n1 ,
nj−1 , j 3, . . . , 2l 1. It is not difficult to know
C
nj
l l−k1
l
Rαi,ik kj i1
C
2l1
l1−k/2
1 ≤ j ≤ l,
Rγi,2l−k−i3 ,
2.4
i1
2l1
l1−k/2
Rγi,2l−k−i3 ,
kj
Rγ2l−k−i3,i
kj il2−k1 /2
kl1
nj
l1
l 1 ≤ j ≤ 2l 1 l ≥ 2,
i1
C
nj
C
0,
C
2l 2 ≤ j.
C
C
C
It is easy to check that nm , nl ⊆ nml for m l ≤ 2l 1 or nm , nl 0 for m l ≥
C
C
C
C
C
C
C
2l 2. For any ψ ∈ Autn0 , we have ψn1 ψn0 , ψn0 n0 , n0 n1 and
C
C
C
C
C
C
ψnj nj , j 2, . . . , 2l 1. Therefore, ψnj−1 \ nj nj−1 \ nj , j 1, . . . , 2l 1. Note
C
C
that if γij ∈ Wn , then γij ∈ n2l−i−j3 \ n2l−i−j4 .
For any maximal ideal M of R, R R/M is a field. The natural homomorphism π :
C
C
R → R induces a homomorphism ψM : tl1 R → tl1 R which is surjective. So every
4
International Journal of Mathematics and Mathematical Sciences
C
C
automorphism ψ of tl1 R may induce an automorphism ψ of tl1 R. Using this fact and
C
that n2l1 Rγ11 for l ≥ 1, we have that ψγ11 c11 γ11 , where c11 ∈ R∗ . Otherwise, c11
C
should be contained in a maximal ideal M of R, then ψγ 11 0 on tl1 R, where γ 11 is the
C
image of γ11 in tl1 R, which is impossible.
C
Lemma 2.2. Let ψ be in Autn0 . If ψαjj , ψαj,j1 and ψγl1,l1 are expressed, respectively, as
l1
j
C
ψ αjj aii αii mod n1 ,
2.5
j 1, . . . , l 1,
i1
l
j
j
C
ψ αj,j1 ai,i1 αi,i1 cl1,l1 γl1,l1 mod n2 ,
2.6
j 1, . . . , l,
i1
l
l1
l1
C
ψ γl1,l1 ai,i1 αi,i1 cl1,l1 γl1,l1 mod n2 ,
2.7
i1
then the following matrices are invertible.
j
i A aji l1×l1 , where aji aii , j 1, . . . , l 1, i 1, . . . , l 1;
j
j
ii B bji l1×l1 , where bji ai,i1 , j 1, . . . , l, i 1, . . . , l, bj,l1 cl1,l1 , j 1, . . . , l,
l1
l1
bl1,i ai,i1 , i 1, . . . , l and bl1,l1 cl1,l1 .
Proof. i That A is invertible follows from the fact that ψ induces an automorphism of the
C
C
C
free R-module n0 /n1 of rank l 1 on the basis {αjj n1 | j 1, . . . , l 1}. ii Note that
C
C
ψ induces an automorphism of the free R-module n1 /n2 of rank l 1 on the basis {αj,j1 C
C
n2 , γl1,l1 n2 | j 1, . . . , l}.
C
Lemma 2.3. Let ψ ∈ Autn0 l ≥ 2. Write ψαjj , ψαj,j1 , and ψγl1,l1 as in 2.5–2.7,
respectively. Then the following conclusions hold.
m
m
m
m
l1
l1
k, ah,h1 cl1,l1 0, ah,h1 ak,k1 0 h /
k
i For 1 ≤ m, k, h ≤ l, ah,h1 ak,k1 0 h /
l1
l1
and ah,h1 cl1,l1 0.
i
i
i
i
i
i
i
k and ahh −ah1,h1 al1,l1 0
ii For 1 ≤ k, h ≤ l, ahh −ah1,h1 akk −ak1,k1 0 h /
1 ≤ h ≤ l, here l ≥ 1, where i 1, l 1.
m
iii For 2 ≤ m ≤ l and 1 ≤ i, k, h ≤ l, aii
m
i /
h/
k/
i, here l ≥ 3 and aii
m
m
m
m
m
− ai1,i1 ahh − ah1,h1 akk − ak1,k1 0
m
m
m
m
− ai1,i1 ahh − ah1,h1 al1,l1 0 1 ≤ i /
h ≤ l.
Proof. i When j / m, m 1, ψαjj , ψαm,m1 0. So
j
j
m
ai,i1 aii − ai1,i1 0,
m
j
cl1,l1 al1,l1 0.
2.8
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5
From ψαmm , ψαm,m1 ψαm,m1 and ψαm1,m1 , ψαm,m1 −ψαm,m1 , we have
m
m
m
m
ai,i1 aii − ai1,i1 ai,i1 ,
m
m
m
2
cl1,l1 al1,l1 cl1,l1 ,
m
m1
m1
m
ai,i1 ,
− ai1,i1 −
ai,i1 aii
m
m1
2.9
m
2
cl1,l1 al1,l1 −
cl1,l1 .
When j /
l 1, ψαjj , ψγl1,l1 0. So
j
j
l1
ai,i1 aii − ai1,i1 0,
l1
j
cl1,l1 al1,l1 0.
2.10
From ψαl1,l1 , ψγl1,l1 2ψγl1,l1 , we have
l1
l1
l1
l1
ai,i1 ,
ai,i1 aii − ai1,i1 2
j
l1
l1
l1
cl1,l1 al1,l1 cl1,l1 .
2.11
j
j
Let C cji l1×l1 , where cji aii − ai1,i1 , j 1, . . . , l 1, i 1, . . . , l, and cj,l1 2al1,l1 ,
m
m
j 1, . . . , l 1. By Lemma 2.2, det A ∈ R∗ , so det C ∈ R∗ . Investigating ah,h1 ak,k1 det C,
we may find that hth column and kth column are linearly dependent both are the form
m
m
m
m
aj,j1 , 0, . . . , 0t , j h, k by 2.6 and 2.7, so ah,h1 ak,k1 det C 0. Similarly,
0, . . . , 0, aj,j1 , −
m
m
l1
l1
l1
l1
m
ah,h1 cl1,l1 det C 0, ah,h1 ak,k1 det C 0 h / k and ah,h1 cl1,l1 det C 0. Then ah,h1
m
m
m
l1 l1
l1 l1
ak,k1 0 h / k, ah,h1 cl1,l1 0, ah,h1 ak,k1 0 h / k, and ah,h1 cl1,l1 0.
1
1
1
1
1
ii When i 1, from ahh − ah1,h1 akk − ak1,k1 det B 0 h / k, we have ahh −
1
1
1
1
1
1
1
ah1,h1 akk − ak1,k1 0 h / k. Similarly, we have ahh − ah1,h1 all al1,l1 0 1 ≤
h ≤ l. When i l 1, we get the results similarly.
iii The proving process is similar to i and ii.
C
Lemma 2.4. Let ψ ∈ Autn0 . Then
1
C
1
i when l ≥ 1, ψα12 a12 α12 mod n2 , where a12 ∈ R∗ ;
1
C
1
i
C
ii if ψα12 a12 α12 mod n2 , where a12 ∈ R∗ , then ψαi,i1 ai,i1 αi,i1 mod n2
l1
C
i
l1
C
C
and ψγl1,l1 cl1,l1 γl1,l1 mod n2 , where ai,i1 , cl1,l1 ∈ R∗ .
C
C
C
C
Proof. i Noting that αi,i1 , γl1,l1 ∈ n1 \n2 and γ12 ∈ n2l \n2l1 , we have ψα12 ∈ n1 \n2
12
C
C
C
12
∈ R∗ . Using 2.7, from ψα11 ,
12 1
1
12
1
1
ψγ12 ψγ12 , we have c12 a11 a22 c12 , that is, a11 a22 1. Write ψα11 and ψγ11 1
C
C
∗
∗
as ψα11 l1
and ψγ11 c11
γ11 ∈ n2l1 , where c11
∈ R∗ . From 2ψγ11 i1 aii αii mod n1
1 ∗
1
1
1
ψα11 , ψγ11 2a11 c11
γ11 , we have a11 1. Then a22 0. By Lemma 2.3 we have aii −
1
1
1
ai1,i1 0, i 2, . . . , l here l ≥ 2 and al1,l1 0. So aii 0, i 2, . . . , l 1, that is, ψα11 1
C
1
C
1
a11 mod n1 . Then ψα12 ψα11 , ψα12 a12 α12 mod n2 and a12 ∈ R∗ . By Lemma 2.3,
and ψγ12 c12 γ12 mod n2l1 ∈ n2l \ n2l1 , where c12
i holds.
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International Journal of Mathematics and Mathematical Sciences
ii Write ψαj,j1 and ψγl1,l1 as 2.6 and 2.7, respectively. From ψα13 ψα12 ,
1 2
C
C
C
C
C
ψα23 , we have ψα13 a12 a23 α13 mod n3 . Since α13 ∈ n2 \ n3 , ψα13 ∈ n2 \ n3 . So
1 2
2
a12 a23 ∈ R∗ , that is, a23 ∈ R∗ . In general, for m 2, . . . , l, we have
ψα1,m1 m
i
C
C
C
ai,i1 α1,m1 mod nm1 ∈ nm \ nm1 ,
m 1, . . . , l,
i1
ψ γ1,l1
2.12
l
i
l1
C
C
C
ai,i1 cl1,l1 γ1,l1 mod nl2 ∈ nl1 \ nl2 ,
ψα1,l1 , ψ γl1,l1 i1
i
j
l1
here ai,i1 , cl1,l1 should be in R∗ , i 1, . . . , l. By Lemma 2.3 we have that ai,i1 0i / j,
j
l1
cl1,l1 0, j 1, . . . , l, and ai,i1 , i 1, . . . , l. Hence ψαl,l1 and ψγl1,l1 have the required
forms, respectively.
C
3. The Standard Automorphisms of tl1 R
C
Now let us introduce two types of Lie automorphisms of tl1 R.
(i) Inner Automorphisms
C
C
Let r In aαij i / j or r In aγij . It is easy to check that ryr −1 ∈ n0 . The map θr : tl1 R →
C
C
C
tl1 R such that x → rxr −1 , x ∈ n0 , defines an automorphism of n0 , which is called an
. We denote θr
inner automorphism note that r is a symplectic matrix defined by −I0l1 Il1
0
−1
−1
by θaαij , θaγij , respectively. In these cases, we have θaα
θ−aαij , θaγ
θ−aγij , respectively,
ij
ij
i, j, θaαij αk,k1 and that θaαij αii αii − aαij , θaαij αjj αjj aαij , θaαij αkk αkk k /
j, i − 1, θaαij αj,j1 αj,j1 aαi,j1 , θaαij αi−1,i αi−1,i − aαi−1,j , θaγii αii αii − 2aγii ,
αk,k1 k /
C
θaγi,i1 αi,i1 αi,i1 − 2aγii and θaαi,i1 γi,i1 γi,i1 2aγii . All inner automorphisms of tl1 R
generate a subgroup of
C
Autn0 ,
which is denoted by I.
(ii) Diagonal Automorphisms
Let di ∈ R∗ , i 0, 1, . . . , l 1, d diagd1 , . . . , dl1 and D diagd, d−1 d0 . The map λD :
C
C
C
C
tl1 R → tl1 R such that x → DxD−1 , x ∈ n0 , defines an automorphism of tl1 R, which
is called a diagonal automorphism. It is clear that λD λD λDD . So the set of diagonal automorC
C
phisms of tl1 R is a subgroup of Autn0 , which is denoted by D.
4. Lemmas for Main Results
C
Lemma 4.1. Let ψ ∈ Autn0 . The following two statements are equivalent:
(i) ψαj,j1 l1
j
aj,j1 αj,j1
cl1,l1 ∈ R∗ , j 1, . . . , l;
C
mod n2
C
l1
C
j
and ψγl1,l1 cl1,l1 γl1,l1 mod n2 , where aj,j1 ,
(ii) ψαjj αjj mod n1 , j 1, . . . , l 1.
International Journal of Mathematics and Mathematical Sciences
7
1
Proof. i⇒ii. Write ψαjj as in 2.5. By the process of proving Lemma 2.3, we have a12
1
1
1
i
i1
i1
i
i1
i1
i1
i1
a11 − a22 a12 , ai,i1 aii − ai1,i1 −
ai,i1 , ai1,i2 ai1,i1 − ai2,i2 ai1,i2 , i 1, . . . , l − 1
l
l1
and al,l1 all
i1
aii
j
i1
l1
l
l1
l1
l1
1
1
− al1,l1 −
al,l1 , cl1,l1 al1,l1 cl1,l1 . Then we obtain that a11 − a22 1,
i1
i1
l1
− ai1,i1 −1, ai1,i1 − ai2,i2 1, i 1, . . . , l − 1 and al1,l1 1. By Lemma 2.3, we have
j
j.
ajj 1 1 ≤ j ≤ l 1 and aii 0 i /
ii⇒i. Write ψαj,j1 and ψγl1,l1 , respectively, as in 2.6 and 2.7. Then
j
C
ψ αj,j1 ψ αjj , ψ αj,j1 aj,j1 αj,j1 mod n2 ,
j 1, . . . , l,
l1
C
cl1,l1 γl1,l1 mod n2 ,
2ψ γl1,l1 ψαl1,l1 , ψ γl1,l1 2
l1
4.1
C
that is, ψγl1,l1 cl1,l1 γl1,l1 mod n2 . By the method of modularizing a maximal ideal of
j
l1
R to a residue field, we know that aj,j1 , cl1,l1 ∈ R∗ , j 1, . . . , l.
C
C
Lemma 4.2. Let ψ be in Autn0 . If ψαjj αjj mod n1 , then
1
C
ψα11 α11 a12 α12 mod n2 ,
j−1
j
C
ψ αjj αjj − aj−1,j αj−1,j aj,j1 αj,j1 mod n2 ,
l
j 2, . . . , l l ≥ 2,
l1
4.2
C
ψαl1,l1 αl1,l1 − al,l1 αl,l1 cl1,l1 γl1,l1 mod n2 .
Proof. We express ψαjj as
l
j
j
C
ψ αjj αjj ai,i1 αi,i1 cl1,l1 γl1,l1 mod n2 ,
j 1, . . . , l 1.
4.3
j 2, . . . , l l ≥ 2,
4.4
i1
From ψαjj , ψαkk 0 j /
k we have
1
C
ψα11 α11 a12 α12 mod n2 ,
j
j
C
ψ αjj αjj aj−1,j αj−1,j aj,j1 αj,j1 mod n2 ,
l1
l1
C
ψαl1,l1 αl1,l1 al,l1 αl,l1 cl1,l1 γl1,l1 mod n2 ,
j
j1
where aj,j1 aj,j1 0, j 1, . . . , l. Lemma 4.2 is proved.
C
Lemma 4.3. Let ψ be in Autn0 . If every ψαjj is expressed as the form in Lemma 4.2, one may
find an inner automorphism
θ
l
j1
θaj
j,j1 αj,j1
θ2−1 cl1
γ
l1,l1 l1,l1
4.5
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International Journal of Mathematics and Mathematical Sciences
such that
C
θψ αjj αjj mod n2 ,
Proof. Note that θ2−1 cl1
4.6
j 1, . . . , l 1.
l1
γ
l1,l1 l1,l1
αl1,l1 αl1,l1 − cl1,l1 γl1,l1 . Then, by Lemma 4.2, it is not
difficult to prove Lemma 4.3.
C
C
Lemma 4.4. Let ψ be in Autn0 . If ψαjj αjj mod nk , j 1, . . . , l 1 1 ≤ k ≤ l 1, then
j
C
ψ αjj αjj aj,jk αj,jk mod nk1 ,
j 1, . . . , min{k, l − k 1}k ≤ l, l ≥ 1,
j−k
j
C
ψ αjj αjj − aj−k,j αj−k,j aj,jk αj,jk mod nk1 ,
l1
,l ≥ 2 ,
j k 1, . . . , l − k 1 k ≤
2
j−k
j
C
ψ αjj αjj − aj−k,j αj−k,j cj,2l−k−j3 γj,2l−k−j3 mod nk1 ,
l1
k
1 k ≤
,l ≥ 1 ,
j l − k 2, . . . , l −
2
2
j
C
ψ αjj αjj cj,2l−k−j3 γj,2l−k−j3 mod nk1 ,
l1
k
1 1
≤ k, l ≥ 2 ,
j l − k 2, . . . , l −
2
2
j
C
ψ αjj αjj cj,2l−k−j3 γj,2l−k−j3 mod nk1 ,
l
l
j 1 k 1, l ≥ 4, here p even ,
2
2
j−k
j
C
ψ αjj αjj − aj−k,j αj−k,j cj,2l−k−j3 γj,2l−k−j3 mod nk1 ,
k
l
l
1 k 1, l ≥ 4, here p even ,
j 2, . . . , l −
2
2
2
2l−k−j3
C
ψ αjj αjj c2l−k−j3,j γ2l−k−j3,j mod nk1 ,
k
k
2, . . . , k l 2 ≤ k ,l ≥ 3 ,
j l−
2
2
j−k
2l−k−j3
C
ψ αjj αjj − aj−k,j αj−k,j c2l−k−j3,j γ2l−k−j3,j mod nk1 ,
k
k
2 , . . . , l 1 k ≤ p,
≥ 1, l ≥ 2 .
j max k 1, l −
2
2
4.7
Proof. We express ψαjj , j 1, . . . , l 1, as
l−k1
j
ai,ik αi,ik ψ αjj αjj i1
l1
j
C
c2l−k−i3,i γ2l−k−i3,i mod nk1 .
il2−k1/2
4.8
International Journal of Mathematics and Mathematical Sciences
9
When k 1 that is the case in Lemma 4.2. The conclusion follows from repeating the process
of proving Lemma 4.4.
C
Lemma 4.5. Let ψ be in Autn0 . If every ψαjj be expressed as the form in Lemma 4.4, one may
find an inner automorphism
θ
l−k1
j1
l−k/21
θaj
α
j,jk j,jk
jl−k2
θ1δjl −1 cj
γ
j,2l−k−j3 j,2l−k−j3
,
4.9
where
δij 1, i j,
h l 1 − k − 1/2k an odd.
0, i / j,
4.10
Then
C
θψ αjj αjj mod nk1 ,
4.11
i 1, . . . , l 1.
Proof. Apply θ to ψαjj and use Lemma 4.4 to obtain Lemma 4.5.
C
C
Lemma 4.6. Let ψ be in Autn0 . If ψαjj αjj mod nk , j 1, . . . , l1, l1 ≤ k ≤ 2l1l ≥ 1,
then
j
C
ψ αjj αjj cj,2l−k−j3 γj,2l−k−j3 mod nk1 ,
2l−k−j3
C
ψ αjj αjj c2l−k−j3,j γ2l−k−j3,j mod nk1 ,
C
ψ αjj αjj mod nk1 ,
j 1, . . . , l −
j l−
k
1,
2
k
2, . . . , 2l − k 2,
2
4.12
j 2l − k 3, . . . , l 1.
Proof. We express ψαjj , j 1, . . . , l 1, as
l−k/21
j
C
ψ αjj αjj ci,2l−k−i3 γi,2l−k−i3 mod nk1 .
4.13
i1
The process of proving Lemma 4.6 is similar to Lemma 4.2.
C
Lemma 4.7. Let ψ be in Autn0 . If every ψαjj is expressed as the form in Lemma 4.6, one may
find an inner automorphism
θ
l−k/21
j1
θ1δjh −1 cj
γ
j,2l−k−j3 j,2l−k−j3
,
4.14
10
International Journal of Mathematics and Mathematical Sciences
where h l 1 − k − 1/2 (k an odd). Then
C
θψ αjj αjj mod nk1 .
4.15
When k 2l 1, θψαjj αjj , j 1, . . . , l 1.
Proof. It is similar to proving Lemma 4.5.
C
Lemma 4.8. When l ≥ 1, let ψ be in Autn0 . If ψαjj αjj , j 1, . . . , l 1, there exists a diagonal
automorphism λD such that λD ψαj,j1 αj,j1 , i 1, . . . , l, and λD ψγl1,l1 γl1,l1 .
j
C
Proof. By Lemma 4.1 we know that ψαj,j1 aj,j1 αj,j1 mod n2
j
C
l1
l1
and ψγl1,l1 cl1,l1
γl1,l1 mod n2 , where aj,j1 , cl1,l1 ∈ R∗ , j 1, . . . , l. We express ψαj,j1 and ψγl1,l1 ,
respectively, as
l l−k1
j
j
ψ αj,j1 aj,j1 αj,j1 ai,ik αi,ik
k2 i1
l
l1
j
c2l−k−i3,i γ2l−k−i3,i
k2 il2−k1/2
2l1
l1−k/2
kl1
j
ci,2l−k−i3 γi,2l−k−i3 ,
i1
4.16
1 ≤ j ≤ l,
l l−k1
l1
l1
ψ γl1,l1 cl1,l1 γl1,l1 ai,ik αi,ik
k2 i1
l
l1
l1
c2l−k−i3,i γ2l−k−i3,i k2 il2−k1/2
2l1
l1−k/2
kl1
i1
l1
ci,2l−k−i3 γi,2l−k−i3 .
Then
ϕ αj,j1 ϕ αjj , ϕ αj,j1 , ϕ αj1,j1
j
j
aj,j1 αj,j1 − cj,j1 γj,j1 ,
j 1, . . . , l.
4.17
In addition,
j
j
ϕ αj,j1 ϕ αjj , aj,j1 αj,j1 − bj,j1 βj,j1 , ϕ αj1,j1
j
aj,j1 αj,j1
j
cj,j1 γj,j1 ,
j 1, . . . , l.
4.18
International Journal of Mathematics and Mathematical Sciences
11
j
Thus cj,j1 0, j 1, . . . , l. So
j
ψ αj,j1 aj,j1 αj,j1 ,
4.19
j 1, . . . , l.
Furthermore,
2ψ γl1,l1 ψαl1,l1 , ψ γl1,l1
l1
2
cl1,l1 γl1,l1 l−1
i1
l1
ai,l1 αi,l1 l1
4.20
l−1
l1
ci,l1 γi,l1 .
i1
l1
From ψαii , ψγl1,l1 0i / l − 1, we have ai,l1 0 and ci,l1 0, i 1, · · · , l − 1, that is,
l1
ψ γl1,l1 cl1,l1 γl1,l1 .
4.21
j j−i1
l1
2
, where d1 1, dj i2 aj−i1,j−i2 , j 2, . . . , l1.
Let d diagd1 , . . . , dl1 and d0 cl1,l1 dl1
Applying λD to ψαj,j1 , j 1, . . . , l, and ψγl1,l1 , we get the result.
5. Proofs of Main Results
Proof of Theorem 1.1. By Lemmas 4.3, 4.5, 4.7, and 4.8 we have λD θψαjj αjj , j 1, . . . , l 1,
λD θψαj,j1 αj,j1 , j 1, . . . , l, and λD θψγl1,l1 γl1,l1 . Since the set {γl1,l1 , αl1,l1 , αjj ,
C
αj,j1 | j 1, . . . , l} generates tl1 R, we know that λD θψ is the identity automorphism of
C
tl1 R. Hence ψ θ λD−1 . The uniqueness of the decomposition follows from Theorem 1.2.
C
Proof of Theorem 1.2. By the first part of Theorem 1.1 we have Auttl1 R ID. For any x ∈
C
tl1 R
and αij ∈
C
n1
we have
−1 −1 −1
λD θaαij λ−1
D
D x D In aαij D xD In aαij
−1
In aλD αij x In aλD αij
5.1
θaλD αij x.
C
So λD θaαij θaλD αij λD . For γij ∈ n1 , we have λD θcγij θcλD γij λD . Therefore, I ID.
Obviously I ∩ D 1. Then, ID I D.
6. Discussion for l 0
C
C
In this case, t1 R is generated by α11 and γ11 . For any automorphism ψ of t1 R, write
ψα11 and ψγ11 , respectively, as ψα11 a11 α11 c11 γ11 and ψγ11 cγ11 , where c ∈
R∗ . From 2ψγ11 ψα11 , ψγ11 , we have a11 1. Then θ2−1 c11 γ11 ψα11 α11 and
θ2−1 c11 γ11 ψγ11 cγ11 . Also ηc θ2−1 c11 γ11 ψα11 α11 and ηc θ2−1 c11 γ11 ψγ11 γ11 . So ψ θ−2−1 c11 γ11
ηc−1 .
12
International Journal of Mathematics and Mathematical Sciences
Acknowledgments
The research was supported partially by National Natural Science Foundation of China
Grant nos. 10871056 and 10971150 and by Science Research Foundation in Harbin Institute
of Technology Grant no. HITC200708.
References
1 G. Ancochea, “On semi-automorphisms of division algebras,” Annals of Mathematics. Second Series,
vol. 48, pp. 147–153, 1947.
2 K. I. Beidar, M. Brešar, and M. A. Chebotar, “Jordan isomorphisms of triangular matrix algebras over
a connected commutative ring,” Linear Algebra and Its Applications, vol. 312, no. 1–3, pp. 197–201, 2000.
3 M. Brešar, “Jordan mappings of semiprime rings,” Journal of Algebra, vol. 127, no. 1, pp. 218–228, 1989.
4 Y. A. Cao, “Automorphisms of the Lie algebra of strictly upper triangular matrices over certain
commutative rings,” Linear Algebra and its Applications, vol. 329, no. 1–3, pp. 175–187, 2001.
5 L. N. Herstein, “Jordan homomorphisms,” Transactions of the American Mathematical Society, vol. 81,
pp. 331–351, 1956.
6 S. Jøndrup, “Automorphisms of upper triangular matrix rings,” Archiv der Mathematik, vol. 49, no. 6,
pp. 497–502, 1987.
7 F. Kuzucuoglu and V. M. Levchuk, “The automorphism group of certain radical matrix rings,” Journal
of Algebra, vol. 243, no. 2, pp. 473–485, 2001.
8 X. T. Wang and Y. M. Li, “Decomposition of Jordan automorphisms of triangular matrix algebra over
commutative rings,” Glasgow Mathematical Journal, vol. 52, no. 3, pp. 529–536, 2010.
9 D. Ž. Doković, “Automorphisms of the Lie algebra of upper triangular matrices over a connected
commutative ring,” Journal of Algebra, vol. 170, no. 1, pp. 101–110, 1994.
10 Y. A. Cao, “Automorphisms of certain Lie algebras of upper triangular matrices over a commutative
ring,” Journal of Algebra, vol. 189, no. 2, pp. 506–513, 1997.
11 X. T. Wang and H. You, “Decomposition of Lie automorphisms of upper triangular matrix algebra
over commutative rings,” Linear Algebra and its Applications, vol. 419, no. 2-3, pp. 466–474, 2006.
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