Decomposition of Automorphisms of

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Volume 2012, Article ID 242736,12pages 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,[email protected] Received 26 March 2012; Accepted 26 April 2012

Academic Editor: Helge Holden

Copyrightq2012 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.

LetCl1Rbe the 2l1×2l1matrix symplectic Lie algebra over a commutative ringRwith 2 invertible. Thent^C_l1R {^m₀¹_−m^m²^T

1 | m1is anl1 upper triangular matrix,m^T2 m2, over R}is the solvable subalgebra ofCl1R. In this paper, we give an explicit description of the automorphism group oft^C_l1R.

1. Introduction

Classical Lie algebras occupy an important place in matrix algebras. LetRbe a commutative ringR with the identity 1 andR^∗ the group of invertible elements inR. LetMnRbe the R-algebra ofnbynmatrices overRthat has a structure of a Lie algebra overRwith bracket operationx, y xy−yxfor anyx, y∈MnR. The symplectic Lie algebra

Cl1R

X|X 0 Il

−Il 0

0 Il

−Il 0

X^T 0, X∈M2l2R

1.1

is one of classical Lie algebras, whereT denotes the matrix transpose. It is easy to show that the following subalgebra ofCl1Rsuch that

t^C_l1R

m1 m2

0 −m^T₁

|m1 is anl1 upper triangular matrix, m^T₂ m2 1.2

is solvable.

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Lete be the identity matrix inMnRand lete_ijⁿdenotes the matrix inMnRall of whose entries are 0, except thei, jth entry which is 1. Let

α_i,ikeⁿ_i,ik−eⁿ_lik1,li1, i1, . . . , l−k1, k0,1, . . . , l, γi,ikeⁿ_i,lik1e_ik,li1ⁿ , i1, . . . , l−k1, k1, . . . , l,

γiie_i,li1ⁿ , i1, . . . , l1,

1.3

wheren2l1,l≥1. For discussion latter, we rewritet^C_l1Ras

t^C_l1R ^l

k0 l−k1

i1

Rα_i,ik^l

k0 l−k1

i1

Rγ_i,ik. 1.4

Automorphisms of associative algebras have been explored in many articles 1–8.

Encouraged by Dokovi´c9and Cao’s10papers which described the automorphism groups of Lie algebra consisting of all upper triangularn×nmatrices of trace 0 over a connected commutative ring and a commutative ring withninvertible, respectively, in this paper we use similar techniques to those in11to prove that any automorphismψ oft^C_l1Rcan be uniquely expressed asψ θλD, where θ and λD are inner and diagonal automorphisms, respectively, forl≥1 andRis a commutative ring with 2 invertible. We also give an explicit description of the remaining casel0.

Theorem 1.1. For any automorphismψoft^C_l1R l≥1there are unique inner and diagonal auto- morphisms,θandλD, respectively, oft^C_l1Rsuch thatψθλD.

Theorem 1.2. Let I and D be the inner and diagonal automorphism groups, respectively. Then Autt^C_l1R ID, where Autt^C_l1Rdenotes the automorphism group oft^C_l1R.

2. Preliminaries

Let

Pn{αi,ik|i1, . . . , l−k1, k0,1, . . . , l}, Wn

γi,ik|i1, . . . , l−k1, k0,1, . . . , l .

2.1

Then the setPn∪Wnis a basis oft^C_l1R.

Lemma 2.1. LetHnbe the set generated by the set{αjj, α_i,i1, γ_l1,l1|1≤j≤l1,1≤i≤l}, where n2l1. ThenHnt^C_l1R.

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Proof. We only need to show thatt^C_l1R⊆ Hn. It is obvious thatα_i,ik ∈Hn, whenk 0,1.

Whenk2, we have

αi,i2 eⁿ_i,i2−e_li3,li1ⁿ

eⁿ_i,i1−e_li2,li1ⁿ ,

e_i1,i2ⁿ −e_li3,li2ⁿ αi,i1, α_i1,i2∈Hn.

2.2

Assume thatαi,ik−1∈Hn, thenαi,ik αi,ik−1, αik−1,ik∈Hn, that is,Pn⊆Hn. Sinceγl1,l1∈ Hn, for anyγl−k1,l−k2∈Wn, whenk1,

γl,l1eⁿ_l,2l2e_l1,2l1ⁿ

eⁿ_l,l1−eⁿ_2l2,2l1

, eⁿ_l1,2l2

αl,l1, γl1,l1

∈Hn.

2.3

Assume that whenkm−1,γ_{l−m2,l−m3} ∈Hn, then whenk m,γ_{l−m1,l−m2} α_{l−m1,l−m3}, γl−m2,l−m3 ∈ Hn, that is,γi,i1 ∈ Hn,i 1, . . . , l. For anyγi,ik ∈ Hn k ≥ 1, whenk 1, γi,i1∈Hn. Whenk≥2,γi,ik αi,ik−1, γik−1,ik∈Hn. Since 2γii αi,i1, γi,i1∈Hn, and 2 is invertible, we haveγii ∈Hnfor 1≤i≤l. ThusWn⊆Hn. BecausePn∪Wnis a basis oft^C_l1R, we obtaint^C_l1R⊆Hn.

Now, denotet^C_l1R byn^C₀ . Let n^C₁ n^C₀ ,n^C₀ ,n^C₂ n^C₁ ,n^C₁ ,n^C_j n^C₁ , n^C_j−1,j 3, . . . ,2l1. It is not diﬃcult to know

n^C_j ^l

kj l−k1

i1

Rαi,ik^l

kj

l1

il2−k1/2

Rγ2l−k−i3,i

^2l1

kl1

l1−k/2

i1

Rγi,2l−k−i3, 1≤j≤l,

n^C_j ^2l1

kj

l1−k/2

i1

Rγi,2l−k−i3, l1≤j ≤2l1l≥2, n^C_j 0, 2l2≤j.

2.4

It is easy to check thatn^Cm ,n^C_l ⊆n^C_mlforml≤2l1 orn^Cm ,n^C_l 0 forml≥ 2l2. For anyψ ∈ Autn^C₀ , we haveψn^C₁ ψn^C₀ , ψn^C₀ n^C₀ ,n^C₀ n^C₁ and ψn^C_j n^C_j ,j 2, . . . ,2l1. Therefore,ψn^C_j−1\n^C_j n^C_j−1\n^C_j ,j 1, . . . ,2l1. Note that ifγij ∈Wn, thenγij∈n^C_2l−i−j3\n^C_2l−i−j4.

For any maximal idealMofR,R R/Mis a field. The natural homomorphismπ : R → R induces a homomorphism ψM : t^C_l1R → t^C_l1R which is surjective. So every

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automorphismψ oft^C_l1Rmay induce an automorphismψ of t^C_l1R. Using this fact and thatn^C_2l1 Rγ11 for l ≥ 1, we have thatψγ11 c11γ11, wherec11 ∈ R^∗. Otherwise,c11

should be contained in a maximal idealMofR, thenψγ₁₁ 0 ont^C_l1R, whereγ₁₁ is the image ofγ11int^C_l1R, which is impossible.

Lemma 2.2. Letψbe in Autn^C₀ . Ifψαjj,ψα_j,j1andψγ_l1,l1are expressed, respectively, as

ψ αjj

^l1

i1

a^j_ii αiimodn^C₁ , j1, . . . , l1, 2.5

ψ αj,j1

^l

i1

a^j_i,i1αi,i1c^j_l1,l1γl1,l1modn^C₂ , j1, . . . , l, 2.6

ψ γ_l1,l1

^l

i1

a^l1_i,i1α_i,i1c^l1_l1,l1γ_l1,l1modn^C₂ , 2.7

then the following matrices are invertible.

iA aji_l1×l1, whereajia^j_ii ,j1, . . . , l1,i1, . . . , l1;

iiB bji_l1×l1, wherebjia^j_i,i1,j 1, . . . , l,i1, . . . , l,bj,l1 c^j_l1,l1,j 1, . . . , l, bl1,ia^l1_i,i1,i1, . . . , landbl1,l1 c^l1_l1,l1.

Proof. iThatAis invertible follows from the fact thatψ induces an automorphism of the freeR-modulen^C₀ /n^C₁ of rankl1 on the basis{αjjn^C₁ | j 1, . . . , l1}.iiNote that ψinduces an automorphism of the freeR-modulen^C₁ /n^C₂ of rankl1 on the basis{αj,j1 n^C₂ , γ_l1,l1n^C₂ | j1, . . . , l}.

Lemma 2.3. Letψ ∈ Autn^C₀ l ≥ 2. Writeψαjj,ψαj,j1, andψγl1,l1as in2.5–2.7, respectively. Then the following conclusions hold.

iFor 1 ≤ m, k, h≤ l,a^m_h,h1a^m_k,k1 0h /k,a^m_h,h1c^m_l1,l1 0,a^l1_h,h1a^l1_k,k1 0h /k and a^l1_h,h1c_l1,l1^l1 0.

iiFor 1≤k, h≤l,aⁱ_hh−aⁱ_h1,h1aⁱ_kk−aⁱ_k1,k1 0 h /k andaⁱ_hh−aⁱ_h1,h1aⁱ_l1,l10 1≤h≤l, herel≥1, where i1, l1.

iiiFor 2 ≤m ≤land 1 ≤ i, k, h ≤l,a^m_ii −a^m_i1,i1a^m_hh −a^m_h1,h1a^m_kk −a^m_k1,k1 0 i /h /k /i, herel≥3anda^m_ii −a^m_i1,i1a^m_hh −a^m_h1,h1a^m_l1,l101≤i /h≤l.

Proof. iWhenj /m, m1,ψαjj, ψα_m,m1 0. So

a^m_i,i1

a^j_ii −a^j_i1,i1

0, c^m_l1,l1a^j_l1,l10. 2.8

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Fromψαmm, ψαm,m1 ψαm,m1andψαm1,m1, ψαm,m1 −ψαm,m1, we have

a^m_i,i1

a^m_ii −a^m_i1,i1

a^m_i,i1, 2c^m_l1,l1a^m_l1,l1c_l1,l1^m ,

a^m_i,i1

a^m1_ii −a^m1_i1,i1

−a^m_i,i1, 2c^m_l1,l1a^m1_l1,l1−c_l1,l1^m .

2.9

Whenj /l1,ψαjj, ψγ_l1,l1 0. So

a^l1_i,i1

a^j_ii −a^j_i1,i1

0, c^l1_l1,l1a^j_l1,l10. 2.10

Fromψα_l1,l1, ψγ_l1,l1 2ψγ_l1,l1, we have

a^l1_i,i1

a^l1_ii −a^l1_i1,i1

2a^l1_i,i1, c_l1,l1^l1 a^l1_l1,l1c^l1_l1,l1. 2.11

LetC cji_l1×l1, wherecjia^j_ii −a^j_i1,i1,j 1, . . . , l1,i1, . . . , l, andc_j,l1 2a^j_l1,l1, j 1, . . . , l1. By Lemma 2.2, detA ∈ R^∗, so detC ∈ R^∗. Investigating a^m_h,h1a^m_k,k1detC, we may find that hth column and kth column are linearly dependent both are the form 0, . . . ,0,a^m_j,j1,−a^m_j,j1,0, . . . ,0^t, j h, kby2.6and2.7, soa^m_h,h1a^m_k,k1detC0. Similarly,

a^m_h,h1c^m_l1,l1detC 0, a^l1_h,h1a^l1_k,k1detC 0h /k anda^l1_h,h1c^l1_l1,l1detC 0. Then a^m_h,h1

a^m_k,k10h /k,a^m_h,h1c^m_l1,l10,a^l1_h,h1a^l1_k,k10 h /k, anda^l1_h,h1c_l1,l1^l1 0.

iiWheni1, froma¹_hh −a¹_h1,h1a¹_kk −a¹_k1,k1detB 0h /k, we havea¹_hh − a¹_h1,h1a¹_kk −a¹_k1,k1 0 h /k. Similarly, we havea¹_hh −a¹_h1,h1a¹_ll a¹_l1,l1 01 ≤ h≤l. Whenil1, we get the results similarly.

iiiThe proving process is similar toiandii.

Lemma 2.4. Letψ ∈Autn^C₀ . Then

iwhenl≥1,ψα12 a¹₁₂α12modn^C₂ , wherea¹₁₂ ∈R^∗;

iiif ψα12 a¹₁₂α12modn^C₂ , where a¹₁₂ ∈ R^∗, then ψα_i,i1 aⁱ_i,i1α_i,i1modn^C₂ andψγl1,l1 c^l1_l1,l1γl1,l1modn^C₂ , whereaⁱ_i,i1,c^l1_l1,l1∈R^∗.

Proof. iNoting thatαi,i1, γl1,l1∈n^C₁ \n^C₂ andγ12∈n^C_2l \n^C_2l1, we haveψα12∈n^C₁ \n^C₂ and ψγ12 c¹²₁₂ γ12modn^C_2l1 ∈ n^C_2l \n^C_2l1, where c¹²₁₂ ∈ R^∗. Using 2.7, from ψα11, ψγ12 ψγ12, we havec₁₂¹²a¹₁₁ a¹₂₂ c¹²₁₂ , that is,a¹₁₁ a¹₂₂ 1. Writeψα11andψγ11 asψα11

_l1

i1a¹_ii αii modn^C₁ andψγ11 c₁₁^∗ γ11 ∈n^C_2l1, wherec^∗₁₁ ∈R^∗. From 2ψγ11

ψα11, ψγ11 2a¹₁₁c^∗₁₁γ11, we havea¹₁₁ 1. Thena¹₂₂ 0. ByLemma 2.3we havea¹_ii − a¹_i1,i1 0, i2, . . . , lherel≥2anda¹_l1,l10. Soa¹_ii 0, i2, . . . , l1, that is,ψα11

a¹₁₁ modn^C₁ . Thenψα12 ψα11, ψα12 a¹₁₂α12modn^C₂ anda¹₁₂ ∈R^∗. ByLemma 2.3, iholds.

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iiWriteψαj,j1andψγl1,l1as2.6and2.7, respectively. Fromψα13 ψα12, ψα23, we haveψα13 a¹₁₂a²₂₃α13 modn^C₃ . Sinceα13∈n^C₂ \n^C₃ ,ψα13∈n^C₂ \n^C₃ . So

a¹₁₂a²₂₃ ∈R^∗, that is,a²₂₃ ∈R^∗. In general, form2, . . . , l, we have

ψα1,m1

m i1

aⁱ_i,i1α1,m1modn^C_m1∈n^Cm \n^C_m1, m1, . . . , l,

ψ γ1,l1

ψα1,l1, ψ

γl1,l1 ^l

i1

aⁱ_i,i1c^l1_l1,l1γ1,l1modn^C_l2 ∈n^C_l1 \n^C_l2,

2.12

hereaⁱ_i,i1, c^l1_l1,l1 should be inR^∗,i 1, . . . , l. By Lemma 2.3we have thata^j_i,i1 0i /j,

c^j_l1,l1 0,j 1, . . . , l, anda^l1_i,i1,i1, . . . , l. Henceψα_l,l1andψγ_l1,l1have the required forms, respectively.

3. The Standard Automorphisms of t

^C_l₁

R

Now let us introduce two types of Lie automorphisms oft^C_l1R.

(i) Inner Automorphisms

Letr Inaαiji /jorrInaγij. It is easy to check thatryr⁻¹∈n^C₀ . The mapθr:t^C_l1R → t^C_l1Rsuch thatx → rxr⁻¹,x ∈ n^C₀ , defines an automorphism ofn^C₀ , which is called an inner automorphismnote thatr is a symplectic matrix defined by

0 Il1

−Il1 0

. We denoteθr

byθaαij,θaγij, respectively. In these cases, we have θ_aα⁻¹_ij θ−aαij,θ⁻¹_aγ_ij θ−aγij, respectively, and thatθaαijαii αii−aαij,θaαijαjj αjj aαij,θaαijαkk αkkk /i, j,θaαijα_k,k1 α_k,k1k /j, i−1,θaαijα_j,j1 α_j,j1aα_i,j1,θaαijα_i−1,i α_i−1,i−aα_i−1,j,θaγiiαii αii−2aγii, θaγi,i1αi,i1 αi,i1−2aγiiandθaαi,i1γi,i1 γi,i12aγii. All inner automorphisms oft^C_l1R generate a subgroup of Autn^C₀ , which is denoted byI.

(ii) Diagonal Automorphisms

Letdi ∈ R^∗,i 0,1, . . . , l1,d diagd1, . . . , dl1andD diagd, d⁻¹d0. The mapλD: t^C_l1R → t^C_l1Rsuch thatx →DxD⁻¹,x∈n^C₀ , defines an automorphism oft^C_l1R, which is called a diagonal automorphism. It is clear thatλDλ_Dλ_DD. So the set of diagonal automorphisms oft^C_l1Ris a subgroup of Autn^C₀ , which is denoted byD.

4. Lemmas for Main Results

Lemma 4.1. Letψ ∈Autn^C₀ . The following two statements are equivalent:

(i)ψα_j,j1 a^j_j,j1αj,j1modn^C₂ andψγ_l1,l1 c^l1_l1,l1γl1,l1modn^C₂ , wherea^j_j,j1, c^l1_l1,l1∈R^∗,j 1, . . . , l;

(ii)ψαjj αjj modn^C₁ , j 1, . . . , l1.

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Proof. i⇒ii. Writeψαjjas in2.5. By the process of proving Lemma 2.3, we have a¹₁₂ a¹₁₁ −a¹₂₂ a¹₁₂,aⁱ_i,i1aⁱ¹_ii −aⁱ¹_i1,i1 −aⁱ_i,i1,aⁱ¹_i1,i2aⁱ¹_i1,i1−aⁱ¹_i2,i2 aⁱ¹_i1,i2,i1, . . . , l−1 anda^l_l,l1a^l1_ll −a^l1_l1,l1 −a^l_l,l1,c^l1_l1,l1a^l1_l1,l1 c_l1,l1^l1 . Then we obtain thata¹₁₁ −a¹₂₂ 1, aⁱ¹_ii −aⁱ¹_i1,i1−1,aⁱ¹_i1,i1−aⁱ¹_i2,i21,i1, . . . , l−1 anda^l1_l1,l1 1. ByLemma 2.3, we have a^j_jj 1 1≤j≤l1anda^j_ii 0i /j.

ii⇒i. Writeψα_j,j1andψγ_l1,l1, respectively, as in2.6and2.7. Then ψ

α_j,j1

ψ αjj

, ψ α_j,j1

a^j_j,j1α_j,j1modn^C₂ , j 1, . . . , l, 2ψ

γ_l1,l1

ψα_l1,l1, ψ

γ_l1,l1

2c_l1,l1^l1 γ_l1,l1modn^C₂ ,

4.1

that is,ψγl1,l1 c^l1_l1,l1γl1,l1modn^C₂ . By the method of modularizing a maximal ideal of Rto a residue field, we know thata^j_j,j1,c^l1_l1,l1∈R^∗,j1, . . . , l.

Lemma 4.2. Letψbe in Autn^C₀ . Ifψαjj αjjmodn^C₁ , then ψα11 α11a¹₁₂α12modn^C₂ , ψ

αjj

αjj−a^j−1_j−1,jα_j−1,ja^j_j,j1α_j,j1modn^C₂ , j2, . . . , ll≥2, ψα_l1,l1 α_l1,l1−a^l_l,l1α_l,l1c^l1_l1,l1γ_l1,l1modn^C₂ .

4.2

Proof. We expressψαjjas

ψ αjj

αjj^l

i1

a^j_i,i1αi,i1c^j_l1,l1γl1,l1modn^C₂ , j1, . . . , l1. 4.3

Fromψαjj, ψαkk 0 j /kwe have

ψα11 α11a¹₁₂α12modn^C₂ , ψ

αjj

αjja^j_j−1,jα_j−1,ja^j_j,j1α_j,j1modn^C₂ , j 2, . . . , l l≥2,

ψα_l1,l1 α_l1,l1a^l1_l,l1α_l,l1c_l1,l1^l1 γ_l1,l1 modn^C₂ ,

4.4

wherea^j_j,j1a^j1_j,j1 0, j1, . . . , l.Lemma 4.2is proved.

Lemma 4.3. Letψ be in Autn^C₀ . If everyψαjjis expressed as the form inLemma 4.2, one may find an inner automorphism

θ^l

j1

θ_a^j

j,j1αj,j1θ₂−1c^l1_l1,l1γl1,l1 4.5

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such that

θψ αjj

αjj modn^C₂ , j1, . . . , l1. 4.6

Proof. Note thatθ₂−1c_l1,l1^l1 γl1,l1α_l1,l1 α_l1,l1−c^l1_l1,l1γ_l1,l1. Then, byLemma 4.2, it is not diﬃcult to proveLemma 4.3.

Lemma 4.4. Letψbe in Autn^C₀ . Ifψαjj αjjmodn^C_k , j1, . . . , l1 1≤k≤l1, then

ψ αjj

αjja^j_j,jkαj,jk modn^C_k1 , j 1, . . . ,min{k, l−k1}k≤l, l≥1,

ψ αjj

αjj−a^j−k_j−k,jαj−k,j a^j_j,jkαj,jkmodn^C_k1, jk1, . . . , l−k1

k≤

l1 2

, l≥2

, ψ

αjj

αjj−a^j−k_j−k,jαj−k,jc_{j,2l−k−j3}^j γj,2l−k−j3modn^C_k1, j l−k2, . . . , l−

k 2

1

k≤ l1

2

, l≥1

, ψ

αjj

αjjc^j_{j,2l−k−j3}γj,2l−k−j3modn^C_k1, j l−k2, . . . , l−

k 2

1

1 l1

2

≤k, l≥2

, ψ

αjj

αjjc^j_{j,2l−k−j3}γj,2l−k−j3modn^C_k1, j l

2 1

k l

21, l≥4, herep even

, ψ

αjj

αjj−a^j−k_j−k,jαj−k,jc_{j,2l−k−j3}^j γj,2l−k−j3modn^C_k1, j l

22, . . . , l− k

2

1

k l

2 1, l≥4, herepeven

, ψ

αjj

αjjc^2l−k−j3_{2l−k−j3,j}γ2l−k−j3,j modn^C_k1, jl−

k 2

2, . . . , k

l2≤k k

2

, l≥3

, ψ

αjj

αjj−a^j−k_j−k,jαj−k,jc_{2l−k−j3,j}^2l−k−j3γ2l−k−j3,j modn^C_k1, j max

k1, l− k

2

, . . . , l1

k≤p, k

2

≥1, l≥2

.

4.7

Proof. We expressψαjj, j1, . . . , l1, as

ψ αjj

αjj^l−k1

i1

a^j_i,ikα_i,ik ^l1

il2−k1/2

c^j_{2l−k−i3,i}γ_{2l−k−i3,i}modn^C_k1. 4.8

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Whenk1 that is the case inLemma 4.2. The conclusion follows from repeating the process of provingLemma 4.4.

Lemma 4.5. Letψbe in Autn^C₀ . If everyψαjjbe expressed as the form inLemma 4.4, one may find an inner automorphism

θ^l−k1

j1

θ_a^j

j,jkα_j,jk

l−k/21

jl−k2

θ_1δ

jl⁻¹c^j_{j,2l−k−j3}γ_{j,2l−k−j3}, 4.9

where

δij

1, ij,

0, i /j, hl1−k−1/2k an odd. 4.10

Then

θψ αjj

αjj modn^C_k1, i1, . . . , l1. 4.11

Proof. Applyθtoψαjjand useLemma 4.4to obtainLemma 4.5.

Lemma 4.6. Letψbe in Autn^C₀ . Ifψαjj αjjmodn^C_k , j1, . . . , l1,l1≤k≤2l1l≥1, then

ψ αjj

αjjc^j_{j,2l−k−j3}γ_{j,2l−k−j3}modn^C_k1, j 1, . . . , l− k

2

1,

ψ αjj

αjjc^2l−k−j3_{2l−k−j3,j}γ_{2l−k−j3,j} modn^C_k1, jl− k

2

2, . . . ,2l−k2, ψ

αjj

αjj modn^C_k1, j2l−k3, . . . , l1.

4.12

Proof. We expressψαjj, j1, . . . , l1, as

ψ αjj

αjj^l−k/21

i1

c_{i,2l−k−i3}^j γ_{i,2l−k−i3}modn^C_k1. 4.13

The process of provingLemma 4.6is similar toLemma 4.2.

Lemma 4.7. Letψ be in Autn^C₀ . If everyψαjjis expressed as the form inLemma 4.6, one may find an inner automorphism

θ^l−k/21

j1

θ_1δ

jh⁻¹c^j_{j,2l−k−j3}γj,2l−k−j3, 4.14

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wherehl1−k−1/2 (kan odd). Then

θψ αjj

αjj modn^C_k1. 4.15

Whenk2l1,θψαjj αjj, j 1, . . . , l1.

Proof. It is similar to provingLemma 4.5.

Lemma 4.8. Whenl≥1, letψbe in Autn^C₀ . Ifψαjj αjj, j 1, . . . , l1, there exists a diagonal automorphismλDsuch thatλDψαj,j1 αj,j1,i1, . . . , l, andλDψγl1,l1 γl1,l1.

Proof. By Lemma 4.1we know that ψα_j,j1 a^j_j,j1αj,j1modn^C₂ and ψγ_l1,l1 c^l1_l1,l1 γ_l1,l1modn^C₂ , wherea^j_j,j1,c^l1_l1,l1 ∈ R^∗,j 1, . . . , l. We express ψα_j,j1and ψγ_l1,l1, respectively, as

ψ αj,j1

a^j_j,j1αj,j1^l

k2 l−k1

i1

a^j_i,ikαi,ik

^l

k2

l1 il2−k1/2

c_{2l−k−i3,i}^j γ_{2l−k−i3,i}

^2l1

kl1

l1−k/2

i1

c^j_{i,2l−k−i3}γi,2l−k−i3, 1≤j≤l,

ψ γl1,l1

c^l1_l1,l1γl1,l1^l

k2 l−k1

i1

a^l1_i,ikαi,ik

^l

k2

l1 il2−k1/2

c^l1_{2l−k−i3,i}γ2l−k−i3,i ^2l1

kl1

l1−k/2

i1

c^l1_{i,2l−k−i3}γi,2l−k−i3.

4.16

Then

ϕ αj,j1

ϕ

αjj

, ϕ

αj,j1 , ϕ

αj1,j1

a^j_j,j1αj,j1−c^j_j,j1γj,j1, j 1, . . . , l.

4.17

In addition,

ϕ αj,j1

ϕ

αjj

,

a^j_j,j1αj,j1−b^j_j,j1βj,j1

, ϕ

αj1,j1

a^j_j,j1α_j,j1c^j_j,j1γ_j,j1, j1, . . . , l.

4.18

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Thusc^j_j,j10, j1, . . . , l. So ψ

αj,j1

a^j_j,j1αj,j1, j1, . . . , l. 4.19 Furthermore,

2ψ γ_l1,l1

ψα_l1,l1, ψ

γ_l1,l1

2c^l1_l1,l1γl1,l1^l−1

i1

a^l1_i,l1αi,l1^l−1

i1

c^l1_i,l1γi,l1. 4.20

Fromψαii, ψγl1,l1 0i /l−1, we havea^l1_i,l1 0 andc^l1_i,l1 0,i1,· · ·, l−1, that is, ψ

γl1,l1

c_l1,l1^l1 γl1,l1. 4.21

Letddiagd1, . . . , dl1andd0c^l1_l1,l1d²_l1, whered1 1, djj

i2a^j−i1_{j−i1,j−i2},j 2, . . . , l1.

ApplyingλDtoψαj,j1,j 1, . . . , l, andψγl1,l1, we get the result.

5. Proofs of Main Results

Proof ofTheorem 1.1. By Lemmas4.3,4.5,4.7, and4.8we haveλDθψαjj αjj, j1, . . . , l1, λDθψα_j,j1 αj,j1, j 1, . . . , l, andλDθψγ_l1,l1 γl1,l1. Since the set{γ_l1,l1,αl1,l1,αjj, αj,j1 | j 1, . . . , l}generatest^C_l1R, we know that λDθψ is the identity automorphism of t^C_l1R. Henceψ θλD⁻¹. The uniqueness of the decomposition follows fromTheorem 1.2.

Proof ofTheorem 1.2. By the first part ofTheorem 1.1we have Autt^C_l1R ID. For anyx∈ t^C_l1Randαij∈n^C₁ we have

λDθaαijλ⁻¹_Dx D

Inaαij

D⁻¹xD

Inaαij

₋₁ D⁻¹

InaλD

αij

x

InaλD

αij

₋₁ θaλDαijx.

5.1

So λDθaαij θaλDαijλD. Forγij ∈ n^C₁ , we haveλDθcγij θcλDγijλD. Therefore, I ID.

ObviouslyI ∩ D1. Then,IDID.

6. Discussion for l 0

In this case,t^C₁ R is generated byα11 and γ11. For any automorphismψ of t^C₁ R, write ψα11 and ψγ11, respectively, as ψα11 a11α11 c11γ11 and ψγ11 cγ11, where c ∈ R^∗. From 2ψγ11 ψα11, ψγ11, we have a11 1. Then θ2⁻¹c11γ11ψα11 α11 and θ2⁻¹c11γ11ψγ11 cγ11. Alsoηcθ2⁻¹c11γ11ψα11 α11 andηcθ2⁻¹c11γ11ψγ11 γ11. Soψ θ₋₂⁻¹c11γ11

ηc⁻¹.

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Acknowledgments

The research was supported partially by National Natural Science Foundation of China Grant nos. 10871056 and 10971150and by Science Research Foundation in Harbin Institute of TechnologyGrant no. HITC200708.

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