Sandwich classification for
O2n+1(R) and U2n+1(R,Δ) revisited

Raimund Preusser

doi:10.1515/jgth-2018-0011

Article Publicly Available

Sandwich classification for O_2n+1(R) and U_2n+1(R,Δ) revisited

Raimund Preusser

Published/Copyright: April 17, 2018

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From the journal Journal of Group Theory Volume 21 Issue 4

Abstract

In a recent paper, the author proved that if n≥3 is a natural number, R a commutative ring and σ∈G⁢Ln⁢(R), then tk⁢l⁢(σi⁢j) where i≠j and k≠l can be expressed as a product of 8 matrices of the form σ±1ε where ε∈En⁢(R). In this article we prove similar results for the odd-dimensional orthogonal groups O2⁢n+1⁢(R) and the odd-dimensional unitary groups U2⁢n+1⁢(R,Δ) under the assumption that R is commutative and n≥3. This yields new, short proofs of the Sandwich Classification Theorems for the groups O2⁢n+1⁢(R) and U2⁢n+1⁢(R,Δ).

1 Introduction

The Sandwich Classification Theorem (SCT) for the lattice of subgroups of a linear group G that are normalized by the elementary subgroup E of G is one of the central points in the structure theory of linear groups. For general rings, not only semilocal or arithmetic ones, the SCT was first proved by H. Bass [4, 5] for G⁢Ln under the stable range condition. This condition allowed him to extract a transvection as well as to prove the standard commutator formula. The first proofs of the SCT for G=G⁢Ln over a commutative ring by J. Wilson [23] and I. Golubchik [7] used only direct calculations as neither localization techniques nor the standard commutator formula were known at that time. About ten years later the SCT was proved applying the standard commutator formula by L. Vaserstein [14] and, independently, by Z. Borewich and N. Vavilov [6]. In the latter article the authors introduced a trick to stabilize a column of a matrix whereas Vaserstein used localization. The next generation of the SCT proofs works for Chevalley groups. L. Vaserstein [15] and E. Abe [1] used localization methods whereas N. Vavilov, E. Plotkin and A. Stepanov [22] introduced the decomposition of unipotents, which in more detail was described in [13] and further developed in [18, 19, 21, 20, 16, 17] by Vavilov and his students and in [9] by V. Petrov. For generalized hyperbolic unitary groups the SCT was announced by A. Bak and N. Vavilov in [3], but the proof has never been published. It appeared later in [10] (which is essentially the author’s thesis). Recently A. Stepanov [12] proved the SCT for all Chevalley groups using the universal localization method.

Let n≥3 be a natural number and R a commutative ring. The SCT for G⁢Ln⁢(R) states the following (it is also true over almost commutative rings, cf. [14]):

SCT.

Let H be a subgroup of G⁢Ln⁢(R). Then H is normalized by En⁢(R) iff

En⁢(R,I)⊆H⊆Cn⁢(R,I)

for some ideal I of R.

The ideal I in the SCT is uniquely determined, namely

I=I⁢(H)={x∈R∣t12⁢(x)∈H}.

Let now σ∈G⁢Ln⁢(R) and set H:=σEn⁢(R), i.e. H is the smallest subgroup of G⁢Ln⁢(R) which contains σ and is normalized by En⁢(R). Then, by the SCT, we have H⊆Cn⁢(R,I) where I=I⁢(H). It follows from the definition of Cn⁢(R,I) that σi⁢j,σi⁢i-σj⁢j∈I for any i≠j. Hence, by the definition of I⁢(H), the matrices t12⁢(σi⁢j) and t12⁢(σi⁢i-σj⁢j) can be expressed as products of matrices of the form σ±1ε where ε∈En⁢(R). In [11] the author showed how one can use the theme of the paper [13] in order to find such expressions and gave boundaries for the number of factors. This yielded a new, very simple proof of the SCT for G⁢Ln⁢(R). Further the author obtained similar results for the even-dimensional orthogonal groups O2⁢n⁢(R) and the even-dimensional unitary groups U2⁢n⁢(R,Λ).

In this article we prove similar results for the odd-dimensional orthogonal groups O2⁢n+1⁢(R) and the odd-dimensional unitary groups U2⁢n+1⁢(R,Δ) under the assumption that R is commutative and n≥3, cf. Theorems 5.2 and 7.2. The proof of the orthogonal version is quite simple. The proof of the unitary version is a bit more complicated, but still it is much shorter than the proof of the SCT for the groups U2⁢n+1⁢(R,Δ) given in [2] (on the other hand, in [2] the ring R is only assumed to be quasi-finite and hence the result is a bit more general). For the odd-dimensional unitary groups U2⁢n+1⁢(R,Δ) this yields the first proof of the SCT which does not use localization.

The rest of the paper is organized as follows. In Section 2 we recall some standard notation which will be used throughout the paper. In Section 3 we state two lemmas which will be used in the proofs of the main theorems 5.2 and 7.2. In Section 4 we recall the definitions of the odd-dimensional orthogonal group O2⁢n+1⁢(R) and some important subgroups. In Section 5 we prove Theorem 5.2. In Section 6 we recall the definitions of the odd-dimensional unitary group U2⁢n+1⁢(R,Δ) and some important subgroups. In Section 7 we prove Theorem 7.2.

2 Notation

By a natural number we mean an element of the set ℕ:={1,2,3,…}. If G is a group and g,h∈G, we let gh:=h⁢g⁢h-1 and [g,h]:=g⁢h⁢g-1⁢h-1. By a ring we will always mean an associative ring with 1 such that 1≠0. Ideal will mean two-sided ideal. If X is a subset of a ring R, then we denote by I⁢(X) the ideal of R generated by X. If X={x}, then we may write I⁢(x) instead of I⁢(X). If n is a natural number and R is a ring, then the set of all n×n matrices with entries in R is denoted by Mn⁢(R). If a∈Mn⁢(R), we denote the entry of a at position (i,j) by ai⁢j, the i-th row of a by ai⁣* and the j-th column of a by a*j. The group of all invertible matrices in Mn⁢(R) is denoted by G⁢Ln⁢(R) and the identity element of G⁢Ln⁢(R) by e. If a∈G⁢Ln⁢(R), then the entry of a-1 at position (i,j) is denoted by ai⁢j′, the i-th row of a-1 by ai⁣*′ and the j-th column of a-1 by a*j′. Further we denote by Rn the set of all row vectors of length n with entries in R and by Rn the set of all column vectors of length n with entries in R. We consider Rn as left R-module and Rn as right R-module. If u∈Rn (resp. u∈Rn), we denote by ut its transpose in Rn (resp. in Rn).

3 Preliminaries

The following two lemmas are easy to check.

Lemma 3.1.

Let G be a group and a,b,c∈G. Then [a,bc]b-1=[b-1,a][a,c].

Lemma 3.2.

Let G be a group, E a subgroup and a∈G. Suppose that b∈G is a product of n elements of the form a±1ε where ε∈E. Then for any ε′∈E

bε′ is a product of n elements of the form a±1ε, and
[ε′,b] is a product of 2⁢n elements of the form a±1ε.

Lemma 3.2 will be used in the proofs of the main theorems without explicit reference.

4 Odd-dimensional orthogonal groups

In this section n denotes a natural number and R a commutative ring. First we recall the definitions of the odd-dimensional orthogonal group O2⁢n+1⁢(R) and the elementary subgroup E⁢O2⁢n+1⁢(R). For an admissible pair (I,J), we recall the definitions of the following subgroups of O2⁢n+1⁢(R); the preelementary subgroup E⁢O2⁢n+1⁢(I,J) of level (I,J), the elementary subgroup E⁢O2⁢n+1⁢(R,I,J) of level (I,J), the principal congruence subgroup O2⁢n+1⁢(R,I,J) of level (I,J), and the full congruence subgroup C⁢O2⁢n+1⁢(R,I,J) of level (I,J).

4.1 The odd-dimensional orthogonal group

Definition 4.1.

Set M:=R2⁢n+1. We use the following indexing for the elements of the standard basis of M: (e1,…,en,e0,e-n,…,e-1). That means that ei is the column whose i-th coordinate is one and all the other coordinates are zero if 1≤i≤n, the column whose (n+1)-st coordinate is one and all the other coordinates are zero if i=0, and the column whose (2⁢n+2+i)-th coordinate is one and all the other coordinates are zero if -n≤i≤-1. We define the quadratic form

Q:M→R,u↦ut⁢(00p010000)⁢u=u1⁢u-1+…+un⁢u-n+u02

where p∈Mn⁢(R) denotes the matrix with ones on the skew diagonal and zeros elsewhere. The subgroup

O2⁢n+1⁢(R):={σ∈G⁢L2⁢n+1⁢(R)∣Q⁢(σ⁢u)=Q⁢(u)⁢ for all ⁢u∈M}

of G⁢L2⁢n+1⁢(R) is called (odd-dimensional) orthogonal group.

Remark 4.2.

The odd-dimensional orthogonal groups O2⁢n+1⁢(R) are special cases of the odd-dimensional unitary groups U2⁢n+1⁢(R,Δ), see [2, Example 15].

Definition 4.3.

Define Θ:={1,…,n,0,-n,…,-1} and Θh⁢b:=Θ∖{0}.

Lemma 4.4.

Let σ∈G⁢L2⁢n+1⁢(R). Then σ∈O2⁢n+1⁢(R) iff the conditions (i) and (ii) below hold:

1. σi⁢j′=σ-j,-i for all i,j∈Θh⁢b,
2. 2⁢σ0⁢j′=σ-j,0 for all j∈Θh⁢b,
3. σi⁢0′=2⁢σ0,-i for all i∈Θh⁢b,
4. 2⁢σ00′=2⁢σ00.
Q⁢(σ*j)=δ0⁢j for all j∈Θ.

Proof.

See [2, Lemma 17]. ∎

4.2 The polarity map

Definition 4.5.

The map

~:M→Mt,u↦(u-1…u-n2⁢u0un…u1)

where Mt=R2⁢n+1 is called the polarity map. Clearly ~ is linear, i.e. u+v~=u~+v~ and u⁢x~=x⁢u~ for any u,v∈M and x∈R.

Lemma 4.6.

If σ∈O2⁢n+1⁢(R) and u∈M, then σ⁢u~=u~⁢σ-1.

Proof.

Follows from Lemma 4.4. ∎

4.3 The elementary subgroup

If i,j∈Θ, let ei⁢j denote the matrix in M2⁢n+1⁢(R) with 1 in the (i,j)-th position and 0 in all other positions.

Definition 4.7.

If i,j∈Θh⁢b, i≠±j and x∈R, the element

Ti⁢j⁢(x):=e+x⁢ei⁢j-x⁢e-j,-i

of O2⁢n+1⁢(R) is called an (elementary) short root matrix. If i∈Θh⁢b and x∈R, the element

Ti⁢(x):=e+x⁢e0,-i-2⁢x⁢ei⁢0-x2⁢ei,-i

of O2⁢n+1⁢(R) is called an (elementary) extra short root matrix. If an element of O2⁢n+1⁢(R) is a short or extra short root matrix, then it is called an elementary matrix. The subgroup of O2⁢n+1⁢(R) generated by all elementary matrices is called the elementary subgroup and is denoted by E⁢O2⁢n+1⁢(R).

Lemma 4.8.

The following relations hold for elementary matrices:

(S1)Ti⁢j⁢(x)=T-j,-i⁢(-x),

(S2)Ti⁢j⁢(x)⁢Ti⁢j⁢(y)=Ti⁢j⁢(x+y),

(S3)[Ti⁢j⁢(x),Tk⁢l⁢(y)]=e if ⁢k≠j,-i⁢ and ⁢l≠i,-j,

(S4)[Ti⁢j⁢(x),Tj⁢k⁢(y)]=Ti⁢k⁢(x⁢y) if ⁢i≠±k,

(S5)[Ti⁢j⁢(x),Tj,-i⁢(y)]=e,

(E1)Ti⁢(x)⁢Ti⁢(y)=Ti⁢(x+y),

(E2)[Ti⁢(x),Tj⁢(y)]=Ti,-j⁢(-2⁢x⁢y) if ⁢i≠±j,

(E3)[Ti⁢(x),Ti⁢(y)]=e,

(SE1)[Ti⁢j⁢(x),Tk⁢(y)]=e if ⁢k≠j,-i,

(SE2)[Ti⁢j⁢(x),Tj⁢(y)]=Tj,-i⁢(-x⁢y2)⁢Ti⁢(x⁢y).

Proof.

Straightforward computation. ∎

Definition 4.9.

Let u∈M be such that u-1=0 and u is isotropic, i.e. Q⁢(u)=0. Then we denote the matrix

(1-u-2…-u-n-2⁢u0-un…-u201u2⋱⋮1un1u01u-n⋱⋮1u-21)

=e+u⁢e-1t-e1⁢u~

=T1⁢(u0)⁢T2,-1⁢(u2)⁢…⁢Tn,-1⁢(un)⁢T-n,-1⁢(u-n)⁢…⁢T-2,-1⁢(u-2)

∈E⁢O2⁢n+1⁢(R)

by T*,-1⁢(u). Clearly T*,-1⁢(u)-1=T*,-1⁢(-u) (note that u~⁢u=0 since u is isotropic) and

(4.1)T*,-1σ⁢(u)=e+σ⁢u⁢σ-1,*′-σ*1⁢u~⁢σ-1=e+σ⁢u⁢σ*1~-σ*1⁢σ⁢u~

for any σ∈O2⁢n+1⁢(R), the last equality by Lemma 4.6.

Definition 4.10.

Let i,j∈Θh⁢b such that i≠±j. Define

Pi⁢j:=e-ei⁢i-ej⁢j-e-i,-i-e-j,-j+ei⁢j-ej⁢i+e-i,-j-e-j,-i

=Ti⁢j⁢(1)⁢Tj⁢i⁢(-1)⁢Ti⁢j⁢(1)∈E⁢O2⁢n+1⁢(R).

It is easy to show that (Pi⁢j)-1=Pj⁢i.

Lemma 4.11.

Let i,j,k∈Θh⁢b such that i≠±j and k≠±i,±j. Let x∈R. Then

Ti⁢jPk⁢i⁢(x)=Tk⁢j⁢(x),
Ti⁢jPk⁢j⁢(x)=Ti⁢k⁢(x), and
TiP-k,-i⁢(x)=Tk⁢(x).

Proof.

Straightforward. ∎

4.4 Relative elementary subgroups

Definition 4.12.

An admissible pair is a pair (I,J) where I and J are ideals of R such that

2⁢J,J2⊆I⊆J

where J2={x2∣x∈J}.

Definition 4.13.

Let (I,J) denote an admissible pair. A short root matrix Ti⁢j⁢(x) is called (I,J)-elementary if x∈I. An extra short root matrix Ti⁢(x) is called (I,J)-elementary if x∈J. If an element of O2⁢n+1⁢(R) is an (I,J)-elementary short or extra short root matrix, then it is called an (I,J)-elementary matrix. The subgroup E⁢O2⁢n+1⁢(I,J) of E⁢O2⁢n+1⁢(R) generated by the (I,J)-elementary matrices is called the preelementary subgroup of level (I,J). Its normal closure E⁢O2⁢n+1⁢(R,I,J) in E⁢O2⁢n+1⁢(R) is called the elementary subgroup of level (I,J).

4.5 Congruence subgroups

In this subsection (I,J) denotes an admissible pair.

Definition 4.14.

The subgroup of O2⁢n+1⁢(R) consisting of all σ∈O2⁢n+1⁢(R) such that

σi⁢j≡δi⁢jmodI for any i∈Θh⁢b,j∈Θ, and
σ0⁢j≡δ0⁢jmodJ for any j∈Θ

is called principal congruence subgroup of level(I,J) and is denoted by O2⁢n+1⁢(R,I,J).

Theorem 4.15.

O2⁢n+1⁢(R,I,J) is a normal subgroup of O2⁢n+1⁢(R).

Proof.

Follows from [2, Corollary 35]. ∎

Definition 4.16.

The subgroup

{σ∈O2⁢n+1⁢(R)∣[σ,E⁢O2⁢n+1⁢(R)]⊆O2⁢n+1⁢(R,I,J)}

of O2⁢n+1⁢(R) is called the full congruence subgroup of level (I,J) and is denoted by C⁢O2⁢n+1⁢(R,I,J).

Lemma 4.17.

Let σ∈O2⁢n+1⁢(R). Then σ∈C⁢O2⁢n+1⁢(R,I,J) iff

σi⁢j∈I for any i∈Θh⁢b,j∈Θ such that i≠j,
σ0⁢j∈J for any j∈Θh⁢b,
σi⁢i-σj⁢j∈I for any i,j∈Θh⁢b, and
σ00-σj⁢j∈J for any j∈Θh⁢b.

Proof.

Straightforward. ∎

Theorem 4.18.

If n≥3, then the following equalities hold:

[C⁢O2⁢n+1⁢(R,I,J),E⁢O2⁢n+1⁢(R)]=[E⁢O2⁢n+1⁢(R,I,J),E⁢O2⁢n+1⁢(R)]

=E⁢O2⁢n+1⁢(R,I,J).

Proof.

See [2, Theorem 39]. ∎

5 Sandwich classification for O2⁢n+1⁢(R)

In this section n denotes a natural number greater than or equal to 3 and R a commutative ring.

Definition 5.1.

Let σ∈O2⁢n+1⁢(R). Then a matrix of the form σ±1ε where ε∈E⁢O2⁢n+1⁢(R) is called an elementary (orthogonal) σ-conjugate.

Theorem 5.2.

Let σ∈O2⁢n+1⁢(R) and i,j,k,l∈Θh⁢b such that i≠±j and k≠±l. Then

Tk⁢l⁢(σi⁢j) is a product of 8 elementary orthogonal σ -conjugates,
Tk⁢l⁢(σi,-i) is a product of 16 elementary orthogonal σ -conjugates,
Tk⁢l⁢(σi⁢0) is a product of 24 elementary orthogonal σ -conjugates,
Tk⁢l⁢(2⁢σ0⁢j) is a product of 24 elementary orthogonal σ -conjugates,
Tk⁢l⁢(σi⁢i-σj⁢j) is a product of 24 elementary orthogonal σ -conjugates,
Tk⁢l⁢(σi⁢i-σ-i,-i) is a product of 48 elementary orthogonal σ -conjugates,
Tk⁢(σ0⁢j) is a product of 64⁢n+148 elementary orthogonal σ -conjugates, and
Tk⁢(σ00-σj⁢j) is a product of 192⁢n+564 elementary orthogonal σ -conjugates.

Proof.

(i) Set τ:=T21⁢(-σ23)⁢T31⁢(σ22)⁢T2,-3⁢(σ2,-1) and ξ:=τ-1σ. One checks easily that (σ⁢τ-1)2⁣*=σ2⁣* and (τ-1⁢σ-1)*,-2=σ*,-2′. Hence ξ2⁣*=e2t and ξ*,-2=e-2. Set

ζ:=[T32(1),[τ,σ]]τ-1=[T32(1),τξ]τ-1=[τ-1,T32(1)][T32(1),ξ],

the last equality by Lemma 3.1. One checks easily that [τ-1,T32⁢(1)]=T31⁢(-σ23) and [T32⁢(1),ξ]=T-2⁢(x-2)⁢∏i≠0,±2Ti⁢2⁢(xi) for some xi∈R (i≠0,2). Hence ζ=T31⁢(-σ23)⁢T-2⁢(x-2)⁢∏i≠0,±2Ti⁢2⁢(xi). It follows that [T12⁢(1),ζ]=T32⁢(σ23). Hence we have shown

[T12(1),[T32(1),[τ,σ]]τ-1]=T32(σ23).

This implies that T32⁢(σ23) is a product of 8 elementary σ-conjugates. It follows from Lemma 4.11 that Tk⁢l⁢(σ23) is a product of 8 elementary σ-conjugates. Since one can bring σi⁢j to position (2,3) conjugating by monomial matrices from E⁢O2⁢n+1⁢(R) (see Definition 4.10), the assertion of (i) follows.

(ii) Clearly the entry of σTj⁢i⁢(1) at position (j,-i) equals σi,-i+σj,-i. Applying (i) to σTj⁢i⁢(1), we get that Tk⁢l⁢(σi,-i+σj,-i) is a product of 8 elementary σ-conjugates (note that any elementary σTj⁢i⁢(1)-conjugate is also an elementary σ-conjugate). Applying (i) to σ, we get that Tk⁢l⁢(σj,-i) is a product of 8 elementary σ-conjugates. It follows that

Tk⁢l⁢(σi,-i)=Tk⁢l⁢(σi,-i+σj,-i)⁢Tj⁢i⁢(-σj,-i)

is a product of 16 elementary σ-conjugates.

(iii) Clearly the entry of σT-j⁢(-1) at position (i,j) equals σi⁢0+σi⁢j-σi,-j. Applying (i) to σT-j⁢(-1), we get that Tk⁢l⁢(σi⁢0+σi⁢j-σi,-j) is a product of 8 elementary σ-conjugates. Applying (i) to σ, we get that Tk⁢l⁢(-σi⁢j) and Tk⁢l⁢(σi,-j) each are a product of 8 elementary σ-conjugates. It follows that

Tk⁢l⁢(σi⁢0)=Tk⁢l⁢(σi⁢0+σi⁢j-σi,-j)⁢Tk⁢l⁢(-σi⁢j)⁢Tk⁢l⁢(σi,-j)

is a product of 24 elementary σ-conjugates.

(iv) Clearly the entry of σTi⁢(-1) at position (i,j) equals 2⁢σ0⁢j+σi⁢j-σ-i,j. Applying (i) to σTi⁢(-1), we get that Tk⁢l⁢(2⁢σ0⁢j+σi⁢j-σ-i,j) is a product of 8 elementary σ-conjugates. Applying (i) to σ, we get that Tk⁢l⁢(-σi⁢j) and Tk⁢l⁢(σ-i,j) each are a product of 8 elementary σ-conjugates. It follows that

Tk⁢l⁢(2⁢σ0⁢j)=Tk⁢l⁢(2⁢σ0⁢j+σi⁢j-σ-i,j)⁢Tk⁢l⁢(-σi⁢j)⁢Tk⁢l⁢(σ-i,j)

is a product of 24 elementary σ-conjugates.

(v) Clearly the entry of σTj⁢i⁢(1) at position (j,i) equals σi⁢i-σj⁢j+σj⁢i-σi⁢j. Applying (i) to σTj⁢i⁢(1), we get that Tk⁢l⁢(σi⁢i-σj⁢j+σj⁢i-σi⁢j) is a product of 8 elementary σ-conjugates. Applying (i) to σ, we get that Tk⁢l⁢(σi⁢j) and Tk⁢l⁢(-σj⁢i) each are a product of 8 elementary σ-conjugates. It follows that

Tk⁢l⁢(σi⁢i-σj⁢j)=Tk⁢l⁢(σi⁢i-σj⁢j+σj⁢i-σi⁢j)⁢Tk⁢l⁢(σi⁢j)⁢Tk⁢l⁢(-σj⁢i)

is a product of 24 elementary σ-conjugates.

(vi) Follows from (v) since

Tk⁢l⁢(σi⁢i-σ-i,-i)=Tk⁢l⁢(σi⁢i-σj⁢j)⁢Tk⁢l⁢(σj⁢j-σ-i,-i).

(vii) Set m:=8. In Step 1 we show that for any x∈R the matrix Tk⁢(x⁢σ0⁢j⁢σj⁢j) is a product of (2⁢n+9)⁢m+4 elementary σ-conjugates. In Step 2 we use Step 1 in order to prove (v).

Step 1. Set

u′:=(0…0σ-1,-1′-σ-1,-2′)t=(0…0σ11-σ21)t∈M

and u:=σ-1⁢u′∈M. Then clearly u-1=0. Further Q⁢(u)=Q⁢(σ-1⁢u′)=Q⁢(u′)=0 and hence u is isotropic. Set

ξ:=T*,-1σ⁢(-u)=e-σ⁢u⁢σ*1~+σ*1⁢σ⁢u~=e-u′⁢σ*1~+σ*1⁢u′~,

the first equality being a consequence of (4.1). Then

ξ=(1-σ11⁢σ21σ11⁢σ11-σ21⁢σ211+σ21⁢σ11-σ31⁢σ21σ31⁢σ111-σ41⁢σ21σ41⁢σ111⋮⋮⋱-σn⁢1⁢σ21σn⁢1⁢σ111-σ01⁢σ21σ01⁢σ111-σ-n,1⁢σ21σ-n,1⁢σ111⋮⋮⋱-σ-4,1⁢σ21σ-4,1⁢σ111-σ-3,1⁢σ21σ-3,1⁢σ111-α0**…***…*-σ31⁢σ11**0α**…***…*σ31⁢σ21**)

where α=σ-1,1⁢σ11+σ-2,1⁢σ21. Set

τ:=T-3,1⁢(σ-3,1⁢σ21)⁢T-3,2⁢(-σ-3,1⁢σ11).

It follows from (i) that τ is a product of 2⁢m elementary σ-conjugates. Clearly

ξ⁢τ=(1-σ11⁢σ21σ11⁢σ11-σ21⁢σ211+σ21⁢σ11-σ31⁢σ21σ31⁢σ111-σ41⁢σ21σ41⁢σ111⋮⋮⋱-σn⁢1⁢σ21σn⁢1⁢σ111-σ01⁢σ21σ01⁢σ111-σ-n,1⁢σ21σ-n,1⁢σ111⋮⋮⋱-σ-4,1⁢σ21σ-4,1⁢σ111001-α-βδ0*…***…****γα-β0*…***…****)

where β=σ-3,1⁢σ21⁢σ31⁢σ11, γ=σ-3,1⁢σ21⁢σ31⁢σ21 and δ=σ-3,1⁢σ11⁢σ31⁢σ11. Let x∈R and set

ζ:=[T2,-3(-x),[T*,-1(u),σ]τ]T*,-1⁢(-u)

=[T2,-3(-x),T*,-1(u)ξτ]T*,-1⁢(-u)

=[T*,-1⁢(-u),T2,-3⁢(-x)]⁢[T2,-3⁢(-x),ξ⁢τ],

the last equality by Lemma 3.1. Clearly ζ is a product of 4⁢m+4 elementary σ-conjugates. One checks easily that

[T*,-1⁢(-u),T2,-3⁢(-x)]

=T1,-2⁢(x⁢u-3)⁢T1,-3⁢(-x⁢u-2)

=T1,-2⁢(x⁢(σ23⁢σ11-σ13⁢σ21))⁢T1,-3⁢(-x⁢(σ22⁢σ11-σ12⁢σ21)).

Further

[T2,-3⁢(-x),ξ⁢τ]

=(∏p=1,p≠3,0-4Tp,-3⁢(x⁢σp⁢1⁢σ11))⁢T-2,-3⁢(x⁢δ)⁢T-1,-3⁢(x⁢(α-β))⁢T3⁢(x⁢σ01⁢σ11).

Hence

ζ=T1,-2⁢(x⁢(σ23⁢σ11-σ13⁢σ21))⁢T1,-3⁢(-x⁢((σ22-σ11)⁢σ11-σ12⁢σ21))

⋅(∏p=2,p≠3,0-4Tp,-3⁢(x⁢σp⁢1⁢σ11))⁢T-2,-3⁢(x⁢δ)⁢T-1,-3⁢(x⁢(α-β))⁢T3⁢(x⁢σ01⁢σ11).

It follows from (i), (ii) and (v) that T3⁢(x⁢σ01⁢σ11) is a product of

4⁢m+4+2⁢m+4⁢m+(2⁢n-5)⁢m+m+3⁢m=(2⁢n+9)⁢m+4

elementary σ-conjugates. In view of Definition 4.10 and Lemma 4.11 we get that Tk⁢(x⁢σ0⁢j⁢σj⁢j) is a product of (2⁢n+9)⁢m+4 elementary σ-conjugates.

Step 2. Clearly

Tk⁢(σ0⁢j)=Tk⁢((∑s∈Θσj⁢s′⁢σs⁢j)⁢σ0⁢j)=∏s∈ΘTk⁢(σj⁢s′⁢σs⁢j⁢σ0⁢j).

By (i) and relation (SE2) in Lemma 4.8, Tk⁢(σj⁢s′⁢σs⁢j⁢σ0⁢j) is a product of 3⁢m elementary σ-conjugates if s≠±j,0. By (ii) and relation (SE2), Tk⁢(σj,-j′⁢σ-j,j⁢σ0⁢j) is a product of 3⋅2⁢m=6⁢m elementary σ-conjugates. By (iv) and relation (SE2), Tk⁢(σj⁢0′⁢σ0⁢j⁢σ0⁢j) is a product of 3⋅3⁢m=9⁢m elementary σ-conjugates (note that σj⁢0′ is a multiple of 2 by Lemma 4.4). By Step 1, Tk⁢(σj⁢j′⁢σj⁢j⁢σ0⁢j) is a product of (2⁢n+9)⁢m+4 elementary σ-conjugates. Thus Tk⁢(σ0⁢j) is a product of

(2⁢n-2)⋅3⁢m+6⁢m+9⁢m+(2⁢n+9)⁢m+4=8⁢n⁢m+18⁢m+4=64⁢n+148

elementary σ-conjugates.

(viii) One checks easily that the entry of σT-j⁢(1) at position (0,j) equals σj⁢j-σ00+σ0⁢j-σj⁢0-σ0,-j-σj,-j. Applying (vii) to σT-j⁢(1), we get that Tk⁢(σj⁢j-σ00+σ0⁢j-σj⁢0-σ0,-j-σj,-j) is a product of 64⁢n+148 elementary σ-conjugates. Applying (vii) to σ, we get that Tk⁢(-σ0⁢j) and Tk⁢(σ0,-j) each are a product of 64⁢n+148 elementary σ-conjugates. By (iii) and relation (SE2) in Lemma 4.8, Tk⁢(σj⁢0) is a product of 3⋅24=72 elementary σ-conjugates. By (ii) and relation (SE2) in Lemma 4.8, Tk⁢(σj,-j) is a product of 3⋅16=48 elementary σ-conjugates. It follows that

Tk⁢(σj⁢j-σ00)=Tk⁢(σj⁢j-σ00+σ0⁢j-σj⁢0-σ0,-j-σj,-j)

⋅Tk⁢(-σ0⁢j)⁢Tk⁢(σj⁢0)⁢Tk⁢(σ0,-j)⁢Tk⁢(σj,-j)

is a product of

3⋅(64⁢n+148)+72+48=192⁢n+564

elementary σ-conjugates. ∎

As a corollary we get the Sandwich Classification Theorem for O2⁢n+1⁢(R).

Corollary 5.3.

Let H be a subgroup of O2⁢n+1⁢(R). Then H is normalized by E⁢O2⁢n+1⁢(R) iff

(5.1)E⁢O2⁢n+1⁢(R,I,J)⊆H⊆C⁢O2⁢n+1⁢(R,I,J)

for some admissible pair (I,J).

Proof.

First suppose that H is normalized by E⁢O2⁢n+1⁢(R). Let (I,J) be the admissible pair defined by

I:={x∈R∣T12⁢(x)∈H} and J:={x∈R∣T1⁢(x)∈H}.

Then clearly E⁢O2⁢n+1⁢(R,I,J)⊆H. The inclusion H⊆C⁢O2⁢n+1⁢(R,I,J), i.e. that if σ∈H, then the conditions (i)–(iv) in Lemma 4.17 are satisfied, follows from Theorem 5.2. Suppose now that (5.1) holds for some admissible pair (I,J). Then it follows from the standard commutator formula in Theorem 4.18 that H is normalized by E⁢O2⁢n+1⁢(R). ∎

6 Odd-dimensional unitary groups

We describe Hermitian form rings (R,Δ) and odd form ideals (I,Ω) first, then the odd-dimensional unitary group U2⁢n+1⁢(R,Δ) and its elementary subgroup E⁢U2⁢n+1⁢(R,Δ) over a Hermitian form ring (R,Δ). For an odd form ideal (I,Ω), we recall the definitions of the following subgroups of U2⁢n+1⁢(R,Δ):

the preelementary subgroup E⁢U2⁢n+1⁢(I,Ω) of level (I,Ω),
the elementary subgroup E⁢U2⁢n+1⁢((R,Λ),(I,Ω)) of level (I,Ω),
the principal congruence subgroup U2⁢n+1⁢((R,Λ),(I,Ω)) of level (I,Ω),
the normalized principal congruence subgroup N⁢U2⁢n+1⁢((R,Λ),(I,Ω)) of level (I,Ω), and
the full congruence subgroup C⁢U2⁢n+1⁢((R,Λ),(I,Ω)) of level (I,Ω).

6.1 Hermitian form rings and odd form ideals

First we recall the definitions of a ring with involution with symmetry and a Hermitian ring.

Definition 6.1.

Let R be a ring and

¯:R→R,x↦x¯

an anti-isomorphism of R (i.e. ¯ is bijective, x+y¯=x¯+y¯, x⁢y¯=y¯⁢x¯ for any x,y∈R and 1¯=1). Further let λ∈R such that x¯¯=λ⁢x⁢λ¯ for any x∈R. Then λ is called a symmetry for ¯, the pair (¯,λ) an involution with symmetry and the triple (R,¯,λ) a ring with involution with symmetry. A subset A⊆R is called involution invariant iff x¯∈A for any x∈A. We call a quadruple (R,¯,λ,μ) where (R,¯,λ) is a ring with involution with symmetry and μ∈R such that μ=μ¯⁢λ a Hermitian ring.

Remark 6.2.

Let (R,¯,λ,μ) be a Hermitian ring.

It is easy to show that λ¯=λ-1.
The map
¯:R→R,x↦x¯:=λ¯⁢x¯⁢λ
is the inverse map of ¯. One checks easily that (R,¯,λ¯,μ¯) is a Hermitian ring.

Next we recall the definition of an R∙-module.

Definition 6.3.

If R is a ring, let R∙ denote the underlying set of the ring equipped with the multiplication of the ring, but not the addition of the ring. A (right) R∙-module is a not necessarily abelian group (G,+.) equipped with a map

∘:G×R∙→G,(a,x)↦a∘x

such that the following holds:

a∘0=0 for any a∈G,
a∘1=a for any a∈G,
(a∘x)∘y=a∘(x⁢y) for any a∈G and x,y∈R, and
(+.⁡ab)∘x=+.⁡(a∘x)(b∘x) for any a,b∈G and x∈R.

A left R∙-module is defined analogously. An R-module is canonically an R∙-module, but not conversely. Let G and G′ be R∙-modules. A group homomorphism f:G→G′ satisfying f⁢(a∘x)=f⁢(a)∘x for any a∈G and x∈R is called a homomorphism of R∙-modules. A subgroup H of G which is ∘-stable (i.e. a∘x∈H for any a∈H and x∈R) is called an R∙-submodule. Further, if A⊆G and B⊆R, then we denote by A∘B the subgroup of G generated by {a∘b∣a∈A,b∈B}. We treat ∘ as an operator with higher priority than +..

An important example of an R∙-module is the Heisenberg group, which we define next. The odd form parameters Δ which are used to define the odd-dimensional unitary groups are certain R∙-submodules of the Heisenberg group.

Definition 6.4.

Let (R,¯,λ,μ) be a Hermitian ring. Define the map

+.:(R×R)×(R×R)→R×R,

((x1,y1),(x2,y2))↦+.⁡(x1,y1)(x2,y2):=(x1+x2,y1+y2-x¯1⁢μ⁢x2).

Then (R×R,+.) is a group, which we call the Heisenberg group and denote by ℌ. Equipped with the map

∘:(R×R)×R∙→R×R,

((x,y),a)↦(x,y)∘a:=(x⁢a,a¯⁢y⁢a)

ℌ becomes an R∙-module.

Remark 6.5.

We denote the inverse of an element (x,y)∈ℌ by -.⁡(x,y). One checks easily that -.⁡(x,y)=(-x,-y-x¯⁢μ⁢x) for any (x,y)∈ℌ.

In order to define the odd-dimensional unitary groups we need the notion of a Hermitian form ring.

Definition 6.6.

Let (R,¯,λ,μ) be a Hermitian ring. Let (R,+) have the R∙-module structure defined by x∘a=a¯⁢x⁢a. Define the trace map

tr:ℌ→R,(x,y)↦x¯⁢μ⁢x+y+y¯⁢λ.

One checks easily that tr is a homomorphism of R∙-modules. Set

Δmin:={(0,x-x¯⁢λ)∣x∈R} and Δmax:=ker⁡(tr).

An R∙-submodule Δ of ℌ lying between Δmin and Δmax is called an odd form parameter of (R,¯,λ,μ). Since Δmin and Δmax are R∙-submodules of ℌ, they are respectively the smallest and largest odd form parameters. A pair ((R,¯,λ,μ),Δ) is called a Hermitian form ring. We shall usually abbreviate it by (R,Δ).

Next we define an odd form ideal of a Hermitian form ring.

Definition 6.7.

Let (R,Δ) be a Hermitian form ring and I an involution invariant ideal of R. Set

J⁢(Δ):={y∈R∣∃z∈R:(y,z)∈Δ},

I~:={x∈R∣J⁢(Δ)¯⁢μ⁢x⊆I}.

Obviously I~ and J⁢(Δ) are right ideals of R, and I⊆I~. Further set

ΩminI:=+.⁡{(0,x-x¯⁢λ)∣x∈I}Δ∘I,

ΩmaxI:=Δ∩(I~×I).

An R∙-submodule Ω of ℌ lying between ΩminI and ΩmaxI is called a relative odd form parameter of level I. Since ΩminI and ΩmaxI are R∙-submodules of ℌ, they are respectively the smallest and the largest relative odd form parameters of level I. If Ω is a relative odd form parameter of level I, then (I,Ω) is called an odd form ideal of (R,Δ).

The following lemma is straightforward to check. It will be used in the proof of Theorem 7.2.

Lemma 6.8.

Let (R,¯,λ,μ) be a Hermitian ring, (a,b)∈Δmax, n∈N and x1,…,xn∈R. Then

(a,b)∘∑i=1nxi=+.⁡(+.i=1n⁡(a,b)∘xi)(0,∑i,j=1,i>jnxi⁢b⁢x¯j-xi⁢b⁢x¯j¯⁢λ).

6.2 The odd-dimensional unitary group

Let (R,Δ) be a Hermitian form ring and n∈ℕ. Let M, e1,…,en,e0,e-n,…,e-1 and p be defined as in Definition 4.1. If u∈M, then we call

(u1,…,un,u-n,…,u-1)t∈R2⁢n

the hyperbolic part of u and denote it by uh⁢b. Further we set

u*:=u¯t and uh⁢b*:=u¯h⁢bt.

Define the maps

b:M×M→R,

(u,v)↦u*⁢(00p0μ0p⁢λ00)⁢v=∑i=1nu¯i⁢v-i+u¯0⁢μ⁢v0+∑i=-n-1u¯i⁢λ⁢v-i

and

q:M→ℌ,

u↦(q1⁢(u),q2⁢(u)):=(u0,uh⁢b*⁢(0p00)⁢uh⁢b)=(u0,∑i=1nu¯i⁢u-i).

Lemma 6.9.

b is a λ-Hermitian form, i.e.
1. b is biadditive,
2. b⁢(u⁢x,v⁢y)=x¯⁢b⁢(u,v)⁢y for all u,v∈M, x,y∈R, and
3. b⁢(u,v)=b⁢(v,u)¯⁢λ for all u,v∈M.
The map q has the following properties:
1. q⁢(u⁢x)=q⁢(u)∘x for all u∈M, x∈R,
2. q⁢(u+v)≡+.⁡+.⁡q⁢(u)q⁢(v)(0,b⁢(u,v))modΔmin for all u,v∈M,
3. tr⁡(q⁢(u))=b⁢(u,u) for all u∈M.

Proof.

Straightforward computation. ∎

Definition 6.10.

The group

U2⁢n+1(R,Δ):={σ∈GL2⁢n+1(R)∣b(σu,σv)=b(u,v) for all u,v∈M

and q(σu)≡q(u)modΔ for all u∈M}

is called the odd-dimensional unitary group.

Remark 6.11.

The odd-dimensional unitary groups U2⁢n+1⁢(R,Δ) are isomorphic to Petrov’s odd hyperbolic unitary groups with V0=R (see [8]). Namely U2⁢n+1⁢(R,Δ) is isomorphic to Petrov’s group U2⁢l⁢(R,𝔏) where Petrov’s pseudoinvolution ^ is defined by x^=-x¯⁢λ, V0=R, B0⁢(a,b)=a^⁢1^-1⁢μ⁢b,

𝔏={(x,y)∈R×R∣(x,-y)∈Δ}

and l=n.

Example 6.12.

The groups U2⁢n+1⁢(R,Δ) include as special cases the even-dimensional unitary groups U2⁢n⁢(R,Λ) where n∈ℕ and (R,Λ) is a form ring, the general linear groups G⁢Ln⁢(R) where R is any ring and n≥2, and further the symplectic groups S⁢pn⁢(R) and the orthogonal groups On⁢(R) where R is a commutative ring and n≥2. For details see [2, Example 15].

Definition 6.13.

Define

Θ+:={1,…,n},Θ-:={-n,…,-1},

Θ:=Θ+∪Θ-∪{0},Θh⁢b:=Θ∖{0}

and the map

ε:Θh⁢b→{±1},i↦{1if ⁢i∈Θ+,-1if ⁢i∈Θ-.

Further if i,j∈Θ, we write i<j iff either i∈Θ+∧j=0 or i=0∧j∈Θ- or i,j∈Θ+∧i<j or i,j∈Θ-∧i<j or i∈Θ+∧j∈Θ-.

Lemma 6.14.

Let σ∈G⁢L2⁢n+1⁢(R). Then σ∈U2⁢n+1⁢(R,Δ) iff the conditions (i) and (ii) below hold:

1. σi⁢j′=λ-(ε⁢(i)+1)/2⁢σ¯-j,-i⁢λ(ε⁢(j)+1)/2 for all i,j∈Θh⁢b,
2. μ⁢σ0⁢j′=σ¯-j,0⁢λ(ε⁢(j)+1)/2 for all j∈Θh⁢b,
3. σi⁢0′=λ-(ε⁢(i)+1)/2⁢σ¯0,-i⁢μ for all i∈Θh⁢b,
4. μ⁢σ00′=σ¯00⁢μ.
q⁢(σ*j)≡(δ0⁢j,0)modΔ for all j∈Θ.

Proof.

See [2, Lemma 17]. ∎

6.3 The polarity map

Definition 6.15.

The map

~:M→M*,u↦(u¯-1⁢λ…u¯-n⁢λu¯0⁢μu¯n…u¯1)

where M*=R2⁢n+1 is called the polarity map. Clearly ~ is involutary linear, i.e. u+v~=u~+v~ and u⁢x~=x¯⁢u~ for any u,v∈M and x∈R.

Lemma 6.16.

If σ∈U2⁢n+1⁢(R,Δ) and u∈M, then σ⁢u~=u~⁢σ-1.

Proof.

Follows from Lemma 6.14. ∎

6.4 The elementary subgroup

We introduce the following notation. In Definition 6.4 we defined an R∙-module structure on ℌ. Let (R,¯,λ¯,μ¯) be the Hermitian ring defined in Remark 6.2 (b). Let ℌ¯ denote the Heisenberg group corresponding to (R,¯,λ¯,μ¯). The underlying set of both ℌ and ℌ¯ is R×R. If we replace the Hermitian ring (R,¯,λ,μ) by the Hermitian ring (R,¯,λ¯,μ¯), then we get another R∙-module structure on R×R. We denote the group operation (resp. scalar multiplication) defined by (R,¯,λ,μ) on R×R by +.1 (resp. ∘1) and the group operation (resp. scalar multiplication) defined by (R,¯,λ¯,μ¯) on R×R by +.-1 (resp. ∘-1). Further we set

Δ1:=Δ and Δ-1:={(x,y)∈R×R∣(x,y¯)∈Δ}.

One checks easily that ((R,¯,λ¯,μ¯),Δ-1) is a Hermitian form ring. Analogously, if (I,Ω) is an odd form ideal of (R,Δ), we set

Ω1:=Ω and Ω-1:={(x,y)∈R×R∣(x,y¯)∈Ω}.

One checks easily that (I,Ω-1) is an odd form ideal of (R,Δ-1).

If i,j∈Θ, let ei⁢j denote the matrix in M2⁢n+1⁢(R) with 1 in the (i,j)-th position and 0 in all other positions.

Definition 6.17.

If i,j∈Θh⁢b, i≠±j and x∈R, the element

Ti⁢j⁢(x):=e+x⁢ei⁢j-λ(ε⁢(j)-1)/2⁢x¯⁢λ(1-ε⁢(i))/2⁢e-j,-i

of U2⁢n+1⁢(R,Δ) is called an (elementary) short root matrix. If i∈Θh⁢b and (x,y)∈Δ-ε⁢(i), the element

Ti⁢(x,y):=e+x⁢e0,-i-λ-(1+ε⁢(i))/2⁢x¯⁢μ⁢ei⁢0+y⁢ei,-i

of U2⁢n+1⁢(R,Δ) is called an (elementary) extra short root matrix. The extra short root matrices of the kind

Ti⁢(0,y)=e+y⁢ei,-i

are called (elementary) long root matrices. If an element of U2⁢n+1⁢(R,Δ) is a short or extra short root matrix, then it is called elementary matrix. The subgroup of U2⁢n+1⁢(R,Δ) generated by all elementary matrices is called the elementary subgroup and is denoted by E⁢U2⁢n+1⁢(R,Δ).

Lemma 6.18.

The following relations hold for elementary matrices:

(S1)Ti⁢j⁢(x)=T-j,-i⁢(-λ(ε⁢(j)-1)/2⁢x¯⁢λ(1-ε⁢(i))/2),

(S2)Ti⁢j⁢(x)⁢Ti⁢j⁢(y)=Ti⁢j⁢(x+y),

(S3)[Ti⁢j⁢(x),Tk⁢l⁢(y)]=e if ⁢k≠j,-i⁢ and ⁢l≠i,-j,

(S4)[Ti⁢j⁢(x),Tj⁢k⁢(y)]=Ti⁢k⁢(x⁢y) if ⁢i≠±k,

(S5)[Ti⁢j⁢(x),Tj,-i⁢(y)]=Ti⁢(0,x⁢y-λ(-1-ε⁢(i))/2⁢y¯⁢x¯⁢λ(1-ε⁢(i))/2),

(E1)Ti⁢(x1,y1)⁢Ti⁢(x2,y2)=Ti⁢(+.-ε⁢(i)⁡(x1,y1)(x2,y2)),

(E2)[Ti⁢(x1,y1),Tj⁢(x2,y2)]=Ti,-j⁢(-λ-(1+ε⁢(i))/2⁢x¯1⁢μ⁢x2) if ⁢i≠±j,

(E3)[Ti⁢(x1,y1),Ti⁢(x2,y2)]=Ti⁢(0,-λ-(1+ε⁢(i))/2⁢(x¯1⁢μ⁢x2-x¯2⁢μ⁢x1)),

(SE1)[Ti⁢j⁢(x),Tk⁢(y,z)]=e if ⁢k≠j,-i,

[Ti⁢j⁢(x),Tj⁢(y,z)]=Tj,-i⁢(z⁢λ(ε⁢(j)-1)/2⁢x¯⁢λ(1-ε⁢(i))/2)

(SE2) ⋅Ti(yλ(ε⁢(j)-1)/2x¯λ(1-ε⁢(i))/2,xzλ(ε⁢(j)-1)/2x¯λ(1-ε⁢(i))/2).

Proof.

Straightforward computation. ∎

Definition 6.19.

Let u∈M be such that u-1=0 and u is isotropic, i.e. q⁢(u)∈Δ. Then we denote the matrix

(1-λ¯⁢u¯-2⁢λ…-λ¯⁢u¯-n⁢λ-λ¯⁢u¯0⁢μ-λ¯⁢u¯n…-λ¯⁢u¯2u1-λ¯⁢u¯11u2⋱⋮1un1u01u-n⋱⋮1u-21)

=e+u⁢e-1t-e1⁢λ¯⁢u~

=T1⁢(q1⁢(u),λ¯⁢(q2⁢(u)-u¯1+λ⁢u1))⁢∏i=2,i≠0-2Ti,-1⁢(ui)∈E⁢U2⁢n+1⁢(R,Δ)

by T*,-1⁢(u). Clearly T*,-1⁢(u)-1=T*,-1⁢(-u) (note that u~⁢u=tr⁡(q⁢(u))=0 since u is isotropic) and

(6.1)T*,-1σ⁢(u)=e+σ⁢u⁢σ-1,*′-σ*1⁢λ¯⁢u~⁢σ-1=e+σ⁢u⁢σ*1~-σ*1⁢λ¯⁢σ⁢u~

for any σ∈U2⁢n+1⁢(R,Δ), the last equality by Lemma 6.16.

Definition 6.20.

Let i,j∈Θh⁢b such that i≠±j. Define

Pi⁢j:=e-ei⁢i-ej⁢j-e-i,-i-e-j,-j+ei⁢j-ej⁢i

+λ(ε⁢(i)-ε⁢(j))/2⁢e-i,-j-λ(ε⁢(j)-ε⁢(i))/2⁢e-j,-i

=Ti⁢j⁢(1)⁢Tj⁢i⁢(-1)⁢Ti⁢j⁢(1)∈E⁢U2⁢n+1⁢(R,Δ).

It is easy to show that (Pi⁢j)-1=Pj⁢i.

Lemma 6.21.

Let i,j,k∈Θh⁢b such that i≠±j and k≠±i,±j. Let x∈R and (y,z)∈Δ-ε⁢(i). Then

Ti⁢jPk⁢i⁢(x)=Tk⁢j⁢(x),
Ti⁢jPk⁢j⁢(x)=Ti⁢k⁢(x), and
TiP-k,-i⁢(y,z)=Tk⁢(y,λ(ε⁢(i)-ε⁢(k))/2⁢z).

Proof.

Straightforward. ∎

Lemma 6.22.

Let σ∈U2⁢n+1⁢(R,Δ) and i,j∈Θh⁢b such that i≠±j. Set σ^:=σPi⁢j. Then

q⁢(σ^*i)={q⁢(σ*j) if ⁢ε⁢(i)=ε⁢(j),+.⁡q⁢(σ*j)(0,-σ¯i⁢j⁢σ-i,j+σ¯i⁢j⁢σ-i,j¯⁢λ-σ¯-j,j⁢σj⁢j+σ¯-j,j⁢σj⁢j¯⁢λ)if ⁢ε⁢(i)=1,ε⁢(j)=-1,+.⁡q⁢(σ*j)(0,-σ¯-i,j⁢σi⁢j+σ¯-i,j⁢σi⁢j¯⁢λ-σ¯j⁢j⁢σ-j,j+σ¯j⁢j⁢σ-j,j¯⁢λ)if ⁢ε⁢(i)=-1,ε⁢(j)=1.

Proof.

Straightforward computation. ∎

6.5 Relative elementary subgroups

Definition 6.23.

Let (I,Ω) denote an odd form ideal of (R,Δ). A short root matrix Ti⁢j⁢(x) is called (I,Ω)-elementary if x∈I. An extra short root matrix Ti⁢(x,y) is called (I,Ω)-elementary if (x,y)∈Ω-ε⁢(i). If an element of U2⁢n+1⁢(R,Δ) is an (I,Ω)-elementary short or extra short root matrix, then it is called an (I,Ω)-elementary matrix. The subgroup E⁢U2⁢n+1⁢(I,Ω) of E⁢U2⁢n+1⁢(R,Δ) generated by the (I,Ω)-elementary matrices is called the preelementary subgroup of level (I,Ω). Its normal closure E⁢U2⁢n+1⁢((R,Δ),(I,Ω)) in E⁢U2⁢n+1⁢(R,Δ) is called the elementary subgroup of level (I,Ω).

6.6 Congruence subgroups

In this subsection (I,Ω) denotes an odd form ideal of (R,Δ). If σ∈M2⁢n+1⁢(R), we call the matrix (σi⁢j)i,j∈Θh⁢b∈M2⁢n⁢(R) the hyperbolic part of σ and denote it by σh⁢b. Further we define the submodule M⁢(R,Δ):={u∈M∣u0∈J⁢(Δ)} of M.

Definition 6.24.

The subgroup

U2⁢n+1((R,Δ),(I,Ω)):={σ∈U2⁢n+1(R,Δ)∣σh⁢b≡eh⁢bmodI and

q(σu)≡q(u)modΩ for all u∈M(R,Δ)}

of U2⁢n+1⁢(R,Δ) is called the principal congruence subgroup of level (I,Ω).

Lemma 6.25.

Let σ∈U2⁢n+1⁢(R,Δ). Then σ∈U2⁢n+1⁢((R,Δ),(I,Ω)) iff the conditions (i) and (ii) below hold:

σh⁢b≡eh⁢bmodI.
1. q⁢(σ*j)∈Ω for all j∈Θh⁢b, and
2. (-.⁡q⁢(σ*0)(1,0))∘a∈Ω for all a∈J⁢(Δ).

Proof.

See [2, Lemma 28]. ∎

Remark 6.26.

Let σ∈U2⁢n+1⁢(R,Δ). Define

I0:={x∈R∣x⁢J⁢(Δ)⊆I},I~0:={x∈R∣J⁢(Δ)¯⁢μ⁢x∈I0}.

Then I0 is a left ideal of R and I~0 an additive subgroup of R. If J⁢(Δ)=R then I0=I and I~0=I~. Lemma 6.25 implies that σ∈U2⁢n+1⁢((R,Δ),(I,ΩmaxI)) iff σ≡emodI,I~,I0,I~0, meaning that σh⁢b≡eh⁢bmodI, σ0⁢j∈I~ for any j∈Θh⁢b, σi⁢0∈I0 for any i∈Θh⁢b and σ00-1∈I~0.

Definition 6.27.

The subgroup

N⁢U2⁢n+1⁢((R,Δ),(I,Ω)):=NormalizerU2⁢n+1⁢(R,Δ)⁡(U2⁢n+1⁢((R,Δ),(I,Ω)))

of U2⁢n+1⁢(R,Δ) is called the normalized principal congruence subgroup of level (I,Ω).

Remark 6.28.

In many interesting situations, N⁢U2⁢n+1⁢((R,Δ),(I,Ω)) equals U2⁢n+1⁢(R,Δ), for example the equality holds if Ω=ΩminI or Ω=ΩmaxI. But it can also happen that N⁢U2⁢n+1⁢((R,Δ),(I,Ω))≠U2⁢n+1⁢(R,Δ), see [2, Example 38].
E⁢U2⁢n+1⁢(R,Δ)⊆N⁢U2⁢n+1⁢((R,Δ),(I,Ω)), see [2, Corollary 36].

Definition 6.29.

The subgroup

C⁢U2⁢n+1⁢((R,Δ),(I,Ω)):=

{σ∈N⁢U2⁢n+1⁢((R,Δ),(I,Ω))∣[σ,E⁢U2⁢n+1⁢(R,Δ)]⊆U2⁢n+1⁢((R,Δ),(I,Ω))}

of U2⁢n+1⁢(R,Δ) is called the full congruence subgroup of level (I,Ω).

Lemma 6.30.

Let σ∈U2⁢n+1⁢(R,Δ). Then σ∈C⁢U2⁢n+1⁢((R,Δ),(I,ΩmaxI)) iff

σi⁢j∈I for any i,j∈Θh⁢b such that i≠j,
σi⁢0∈I0 for any i∈Θh⁢b,
σ0⁢j∈I~ for any j∈Θh⁢b,
σi⁢i-σj⁢j∈I for any i,j∈Θh⁢b, and
σ00-σj⁢j∈I~0 for any j∈Θh⁢b.

Theorem 6.31.

If R is semilocal or quasifinite and n≥3, then the following equalities hold:

[C⁢U2⁢n+1⁢((R,Δ),(I,Ω)),E⁢U2⁢n+1⁢(R,Δ)]

=[E⁢U2⁢n+1⁢((R,Δ),(I,Ω)),E⁢U2⁢n+1⁢(R,Δ)]

=E⁢U2⁢n+1⁢((R,Δ),(I,Ω)).

Proof.

See [2, Theorem 39]. ∎

7 Sandwich classification for U2⁢n+1⁢(R,Δ)

In this section n denotes a natural number greater than or equal to 3 and (R,Δ) a Hermitian form ring where R is commutative.

Definition 7.1.

Let σ∈U2⁢n+1⁢(R,Δ). Then a matrix of the form σ±1ε where ε∈E⁢U2⁢n+1⁢(R,Δ) is called an elementary (unitary) σ-conjugate.

Theorem 7.2.

Let σ∈U2⁢n+1⁢(R,Δ) and i,j,k,l∈Θh⁢b such that k≠±l and i≠±j. Further let a∈J⁢(Δ). Then

Tk⁢l⁢(σi⁢j) is a product of 160 elementary unitary σ -conjugates,
Tk⁢l⁢(σi,-i) is a product of 320 elementary unitary σ -conjugates,
Tk⁢l⁢(σi⁢0⁢a) is a product of 480 elementary unitary σ -conjugates,
Tk⁢l⁢(a¯⁢μ⁢σ0⁢j) is a product of 480 elementary unitary σ -conjugates,
Tk⁢l⁢(σi⁢i-σj⁢j) is a product of 480 elementary unitary σ -conjugates,
Tk⁢l⁢(σi⁢i-σ-i,-i) is a product of 960 elementary unitary σ -conjugates,
Tk⁢(q1⁢(σ*j),λ-(ε⁢(k)+1)/2⁢q2⁢(σ*j)) is a product of 1600⁢n+5764 elementary unitary σ -conjugates, and
Tk⁢((σ00-σj⁢j)⁢a,x) is a product of 4800⁢n+16812 elementary unitary σ -conjugates for some x∈R.

Proof.

(i) Let x∈R. In Step 1 we show that Tk⁢l⁢(x⁢σ¯23⁢σ21) is a product of 16 elementary σ-conjugates. In Step 2 we show that Tk⁢l⁢(x⁢σ¯23⁢σ2,-1) is a product of 16 elementary σ-conjugates. In Step 3 we show that Tk⁢l⁢(x⁢σ¯23⁢σ22) is a product of 32 elementary σ-conjugates. In Step 4 we use Steps 1–3 in order to prove (i).

Step 1. Set

τ:=T1,-2⁢(σ¯23⁢σ23)⁢T3,-2⁢(-σ¯23⁢σ21)⁢T3,-1⁢(λ¯⁢σ¯23⁢σ22)⁢T3⁢(0,σ¯22⁢σ21-λ¯⁢σ¯21⁢σ22)

and ξ:=τ-1σ. One checks easily that (σ⁢τ-1)2⁣*=σ2⁣* and (τ-1⁢σ-1)*,-2=σ*,-2′. Hence ξ2⁣*=e2t and ξ*,-2=e-2. Set

ζ:=[T-2,-1(1),[τ,σ]]τ-1

=[T-2,-1(1),τξ]τ-1

=[τ-1,T-2,-1⁢(1)]⁢[T-2,-1⁢(1),ξ],

the last equality by Lemma 3.1. One checks easily that

[τ-1,T-2,-1⁢(1)]=T3,-1⁢(σ¯23⁢σ21)⁢T1⁢(z)

for some z∈Δ-1 and [T-2,-1⁢(1),ξ]=T-2⁢(x-2)⁢∏i≠0,±2Ti⁢2⁢(xi) for some xi∈R (i≠0,2). Hence

ζ=T3,-1⁢(σ¯23⁢σ21)⁢T1⁢(z)⁢T-2⁢(x-2)⁢∏i≠0,±2Ti⁢2⁢(xi).

It follows that [T-1,3⁢(-x),[T-2,3⁢(1),ζ]]=T-2,3⁢(x⁢σ¯23⁢σ21) for any x∈R. Hence we have shown

[T-1,3(-x),[T-2,3(1),[T-2,-1(1),[τ,σ]]τ-1]]=T-2,3(xσ¯23σ21).

This implies that T-2,3⁢(x⁢σ¯23⁢σ21) is a product of 16 elementary σ-conjugates. It follows from Lemma 6.21 that Tk⁢l⁢(x⁢σ¯23⁢σ21) is a product of 16 elementary σ-conjugates.

Step 2. Step 2 can be done similarly. Namely one can show that

[T-1,3(-xλ¯),[T12(1),[T-1,2(1),[τ,σ]]τ-1]]=T-1,2(xσ¯23σ2,-1)

for any x∈R where

τ=T21⁢(σ¯23⁢σ23)⁢T31⁢(-σ¯23⁢σ22)⁢T3,-2⁢(σ¯23⁢σ2,-1)⁢T3⁢(0,-σ¯22⁢σ2,-1+λ¯⁢σ¯2,-1⁢σ22).

Step 3. Set

τ:=T21⁢(-σ¯22⁢σ23)⁢T31⁢(σ¯22⁢σ22)⁢T2,-3⁢(σ¯22⁢σ2,-1)⁢T2⁢(0,-σ¯23⁢σ2,-1+λ¯⁢σ¯2,-1⁢σ23)

and ξ:=τ-1σ. One checks easily that (σ⁢τ-1)2⁣*=σ2⁣* and (τ-1⁢σ-1)*,-2=σ*,-2′. Hence ξ2⁣*=e2t and ξ*,-2=e-2. Set

ζ:=[T32(1),[τ,σ]]τ-1=[T32(1),τξ]τ-1=[τ-1,T32(1)][T32(1),ξ].

One checks easily that ψ:=[τ-1,T32⁢(1)]=T31⁢(-σ¯22⁢σ23)⁢T3⁢(y)⁢T3,-2⁢(z) for some y∈Δ-1 and z∈R and θ:=[T32⁢(1),ξ]=T-2⁢(x-2)⁢∏i≠0,±2Ti⁢2⁢(xi) for some xi∈R (i≠0,2). Set

χ:=[T12(1),ζ]ψ-1=[T12(1),ψθ]ψ-1=[ψ-1,T12(1)][T12(1),θ].

One checks easily that [ψ-1,T12⁢(1)]=T32⁢(σ¯22⁢σ23)⁢T3⁢(a)⁢T3,-1⁢(b) for some a∈Δ-1 and b∈R and [T12⁢(1),θ]=T-2⁢(c) for some c∈Δ. Hence

χ=T32⁢(σ¯22⁢σ23)⁢T3⁢(a)⁢T3,-1⁢(b)⁢T-2⁢(c).

It follows that

[T-2,3⁢(x¯),[T2,-1⁢(1),χ]]=T-2,-1⁢(-x¯⁢σ¯22⁢σ23)=T12⁢(x⁢σ¯23⁢σ22)

for any x∈R, the last equality by (S1) of Lemma 6.18. Hence we have shown

[T-2,3(x¯),[T2,-1(1),[T12(1),[T32(1),[τ,σ]]τ-1]ψ-1]]=T12(xσ¯23σ22).

This implies that T12⁢(x⁢σ¯23⁢σ22) is a product of 32 elementary σ-conjugates. It follows from Lemma 6.21 that Tk⁢l⁢(x⁢σ¯23⁢σ22) is a product of 32 elementary σ-conjugates.

Step 4. Set I:=I⁢({σ¯23⁢σ21,σ¯23⁢σ2,-1¯}), J:=I⁢({σ¯23⁢σ21,σ¯23⁢σ2,-1¯,σ¯23⁢σ22}) and

τ:=[σ-1,T12⁢(-σ¯23)]=(e-σ*1′⁢σ¯23⁢σ2⁣*+σ*,-2′⁢σ23⁢σ-1,*)⁢T12⁢(σ¯23).

Clearly

τ11=1-σ11′⁢σ¯23⁢σ21+σ1,-2′⁢σ23⁢σ-1,1

=1-σ11′⁢σ¯23⁢σ21⏟∈I+λ¯⁢σ¯2,-1⁢σ23⏟∈I⁢σ-1,1,

the last equality by Lemma 6.14, and

τ12=-σ11′⁢σ¯23⁢σ22+σ1,-2′⁢σ23⁢σ-1,2+τ11⁢σ¯23

=-σ11′⁢σ¯23⁢σ22⏟∈J+λ¯⁢σ¯2,-1⁢σ23⏟∈J⁢σ-1,2+τ11⁢σ¯23,

again the last equality by Lemma 6.14. Hence τ11≡1modI and τ12≡σ¯23modJ (note that I⊆J). Set ζ:=τP13⁢P21. Then ζ22=τ11 and ζ23=τ12 and hence ζ¯23⁢ζ22≡σ23modI+J¯. Applying Step 3 above to ζ, we get that Tk⁢l⁢(ζ¯23⁢ζ22) is a product of 32 elementary ζ-conjugates. Since any elementary ζ-conjugate is a product of two elementary σ-conjugates, it follows that Tk⁢l⁢(ζ¯23⁢ζ22) is a product of 64 elementary σ-conjugates. Thus, by Steps 1–3, Tk⁢l⁢(σ23) is a product of 64+16+16+16+16+32=160 elementary σ-conjugates. Since one can bring σi⁢j to position (3,2) conjugating by monomial matrices from E⁢U2⁢n+1⁢(R,Δ), the assertion of (i) follows.

(ii)–(vi) See the proof of Theorem 5.2. In the proof of (iii) replace the matrix T-j⁢(-1) by T-j⁢(-.ε⁢(j)⁡(a,b)) where b is chosen such that (a,b)∈Δε⁢(j). In the proof of (iv) replace the matrix Ti⁢(-1) by Ti⁢(-.-ε⁢(i)⁡(λ(ε⁢(i)-1)/2⁢a,b)) where b is chosen such that (a,b)∈Δ-ε⁢(i).

(vii) By Lemma 6.21 it suffices to consider the case ε⁢(k)=-1. Set m:=160. In Step 1 we show that for any x∈R the matrix Tk⁢(q⁢(σ*1)∘σ11⁢x) is a product of (2⁢n+19)⁢m+4 elementary σ-conjugates. In Step 2 we use Step 1 in order to prove (vii).

Step 1. Set

u′:=(0…0σ-1,-1′-σ-1,-2′)t=(0…0σ¯11-σ¯21)t∈M

and u:=σ-1⁢u′∈M. Then clearly u-1=0. Further

q⁢(u)=q⁢(σ-1⁢u′)≡q⁢(u′)=(0,0)modΔ

and hence u is isotropic. Set

ξ:=T*,-1σ⁢(-u)=e-σ⁢u⁢σ*1~+σ*1⁢λ¯⁢σ⁢u~=e-u′⁢σ*1~+σ*1⁢λ¯⁢u′~,

the first equality being a consequence of (6.1). Then

ξ=(1-σ11⁢σ21σ11⁢σ11-σ21⁢σ211+σ21⁢σ11-σ31⁢σ21σ31⁢σ111-σ41⁢σ21σ41⁢σ111⋮⋮⋱-σn⁢1⁢σ21σn⁢1⁢σ111-σ01⁢σ21σ01⁢σ111-σ-n,1⁢σ21σ-n,1⁢σ111⋮⋮⋱-σ-4,1⁢σ21σ-4,1⁢σ111-σ-3,1⁢σ21σ-3,1⁢σ111-λ⁢β¯α**…***…*-σ¯11⁢σ¯31**γβ**…***…*σ¯21⁢σ¯31**)

where

α=σ-2,1⁢σ11-λ⁢σ¯11⁢σ¯-2,1,

β=σ-1,1⁢σ11+λ⁢σ¯21⁢σ¯-2,1,

γ=-σ-1,1⁢σ21+λ⁢σ¯21⁢σ¯-1,1.

Set

τ:=T-3,1⁢(σ-3,1⁢σ21)⁢T-3,2⁢(-σ-3,1⁢σ11).

It follows from (i) that τ is a product of 2⁢m elementary σ-conjugates. Clearly

where δ=σ¯11⁢σ¯31⁢σ-3,1⁢σ11 and ε=σ¯21⁢σ¯31⁢σ-3,1⁢σ11. Let x∈R and set

ζ:=[T2,-3(-x),[T*,-1(u),σ]τ]T*,-1⁢(-u)

=[T2,-3(-x),T*,-1(u)ξτ]T*,-1⁢(-u)

=[T*,-1⁢(-u),T2,-3⁢(-x)]⁢[T2,-3⁢(-u),ξ⁢τ].

Clearly ζ is a product of 4⁢m+4 elementary σ-conjugates. One checks easily that

[T*,-1⁢(-u),T2,-3⁢(-x)]

=T1⁢(0,λ¯⁢(-x¯⁢u-2⁢u¯-3+λ⁢x¯⁢u-2⁢u¯-3¯))⁢T1,-2⁢(λ¯⁢x¯⁢u¯-3)⁢T1,-3⁢(-x⁢u¯-2)

=T1⁢(0,λ¯⁢(a+λ⁢a¯))⁢T1,-2⁢(b)⁢T1,-3⁢(-x⁢(σ11⁢σ22-σ12⁢σ21))

for some a,b∈I⁢({σ21,σ23}). Further

[T2,-3⁢(-x),ξ⁢τ]

=(∏p=1,p≠3,0-4Tp,-3⁢(x⁢σp⁢1⁢σ11))⁢T-2,-3⁢(x⁢(α+δ))⁢T-1,-3⁢(x⁢(β-ε))⁢T3⁢(y)

where

y=(q1⁢(σ*1),λ¯⁢q2⁢(σ*1))∘σ11⁢x+(0,c-λ¯⁢c¯)

for some c∈I⁢({σ31,σ-2,1}). Hence

ζ=T1⁢(0,λ¯⁢(a+λ⁢a¯))⁢T1,-2⁢(b)⁢T1,-3⁢(-x⁢(σ11⁢(σ22-σ11)-σ12⁢σ21))

⋅(∏p=2,p≠3,0-4Tp,-3⁢(x⁢σp⁢1⁢σ11))⁢T-2,-3⁢(x⁢(α+δ))⁢T-1,-3⁢(x⁢(β-ε))⁢T3⁢(y).

It follows from (i), (ii) and (iii) and relation (S5) in Lemma 6.18 that T3⁢(y) is a product of

4⁢m+4+4⁢m+2⁢m+4⁢m+(2⁢n-5)⁢m+3⁢m+3⁢m=(2⁢n+15)⁢m+4

elementary σ-conjugates. By (i) and relation (S5) in Lemma 6.18, T3⁢(0,-c+λ¯⁢c¯) is a product of 4⁢m elementary σ-conjugates. Hence

T3⁢((q1⁢(σ*1),λ¯⁢q2⁢(σ*1))∘σ11⁢x)=T3⁢(y)⁢T3⁢(0,-c+λ¯⁢c¯)

is a product of (2⁢n+19)⁢m+4 elementary σ-conjugates. It follows from Lemma 6.21 that Tk⁢(q⁢(σ*1)∘σ11⁢x) is a product of (2⁢n+19)⁢m+4 elementary σ-conjugates.

Step 2. By Lemma 6.8 we have

Tk⁢(q⁢(σ*1))=Tk⁢(q⁢(σ*1)∘∑s=1-1σ1⁢s′⁢σs⁢1)

=Tk((+.s=1-1(q(σ*1)∘σ1⁢s′σs⁢1))

+.(0,∑s,t=1,s>t-1σ1⁢s′σs⁢1q2(σ*1)σ1⁢t′⁢σt⁢1¯-σ1⁢s′⁢σs⁢1⁢q2⁢(σ*1)⁢σ1⁢t′⁢σt⁢1¯¯λ))

=∏s=1-1Tk⁢(q⁢(σ*1)∘σ1⁢s′⁢σs⁢1)⏟A:=

⋅Tk⁢(0,∑s,t=1,s>t-1σ1⁢s′⁢σs⁢1⁢q2⁢(σ*1)⁢σ1⁢t′⁢σt⁢1¯-σ1⁢s′⁢σs⁢1⁢q2⁢(σ*1)⁢σ1⁢t′⁢σt⁢1¯¯⁢λ)⏟B:=

since q⁢(σ*1)∈Δ⊆Δmax. By Step 1, Tk⁢(q⁢(σ*1)∘σ11′⁢σ11) is a product of (2⁢n+19)⁢m+4 elementary σ-conjugates. By (i) and (SE2) in Lemma 6.18, Tk⁢(q⁢(σ*1)∘σ1⁢s′⁢σs⁢1) is a product of 3⁢m elementary σ-conjugates if s≠±1,0. By (ii) and again (SE2), Tk⁢(q⁢(σ*1)∘σ1,-1′⁢σ-1,1) is a product of 6⁢m elementary σ-conjugates. By (iv) and (SE2), Tk⁢(q⁢(σ*1)∘σ10′⁢σ01) is a product of 9⁢m elementary σ-conjugates (note that σ10′=λ¯⁢σ¯0,-1⁢μ by Lemma 6.14). Hence A is a product of

(2⁢n+19)⁢m+4+(2⁢n-2)⋅3⁢m+6⁢m+9⁢m=(8⁢n+28)⁢m+4

elementary σ-conjugates. On the other hand B=Tk⁢(0,x-λ⁢x¯) for some x∈I⁢(q2⁢(σ*1)). Since q2⁢(σ*1)=∑r=1nσ¯r⁢1⁢σ-r,1, it follows from (i), (ii) and relation (S5) in Lemma 6.18 that B is a product of

4⁢m+(n-1)⋅2⁢m=(2⁢n+2)⁢m

elementary σ-conjugates. Hence Tk⁢(q⁢(σ*1)) is a product of (10⁢n+30)⁢m+4=1600⁢n+4804 elementary σ-conjugates. The assertion of (vii) follows now from Lemma 6.22.

(viii) By Lemma 6.21 it suffices to consider the case ε⁢(k)=-1. Choose a b∈R such that (a,b)∈Δε⁢(j). One checks easily that the entry of ξ:=σT-j⁢(a,b) at position (0,j) equals (σj⁢j-σ00)⁢a+σ0⁢j-σj⁢0⁢a2+σ0,-j⁢b+σj,-j⁢b⁢a2. Applying (vii) to ξ, we get that

Tk⁢(q⁢(ξ*j))=Tk⁢((σj⁢j-σ00)⁢a+σ0⁢j-σj⁢0⁢a2+σ0,-j⁢b+σj,-j⁢b⁢a2,x1)

is a product of 1600⁢n+4804 elementary σ-conjugates where x1 is some ring element. Applying (vii) to σ, we get that

Tk⁢(-.⁡q⁢(σ*j))=Tk⁢(-σ0⁢j,x2) and Tk⁢(-.⁡q⁢(σ*,-j)∘b)=Tk⁢(-σ0,-j⁢b,x3)

each are a product of 1600⁢n+4804 elementary σ-conjugates where x2,x3 are some ring elements. By (iii) and (SE2) in Lemma 6.18, Tk⁢(σj⁢0⁢a2,x4) is a product of 3⋅480=1440 elementary σ-conjugates where x4 is some ring element. By (ii) and again (SE2), Tk⁢(-σj,-j⁢b⁢a2,x5) is a product of 3⋅320=960 elementary σ-conjugates where x5 is some ring element. It follows that

Tk⁢((σj⁢j-σ00)⁢a+σ0⁢j-σj⁢0⁢a2+σ0,-j⁢b+σj,-j⁢b⁢a2,x1)

⋅Tk⁢(-σ0⁢j,x2)⁢Tk⁢(σj⁢0⁢a2,x4)⁢Tk⁢(-σ0,-j⁢b,x3)⁢Tk⁢(-σj,-j⁢b⁢a2,x5)

=Tk⁢((σj⁢j-σ00)⁢a,x)

is a product of 3⋅(1600⁢n+4804)+1440+960=4800⁢n+16812 elementary σ-conjugates where x is some ring element. ∎

As a corollary we get the Sandwich Classification Theorem for U2⁢n+1⁢(R,Δ).

Corollary 7.3.

Let H be a subgroup of U2⁢n+1⁢(R,Δ). Then H is normalized by E⁢U2⁢n+1⁢(R,Δ) iff

(7.1)E⁢U2⁢n+1⁢((R,Δ),(I,Ω))⊆H⊆C⁢U2⁢n+1⁢((R,Δ),(I,Ω))

for some odd form ideal (I,Ω) of (R,Δ).

Proof.

First suppose that H is normalized by E⁢U2⁢n+1⁢(R,Δ). Let (I,Ω) be the odd form ideal of (R,Δ) defined by

I:={x∈R∣T12⁢(x)∈H} and Ω:={(y,z)∈Δ∣T-1⁢(y,z)∈H}.

So the first inclusion of (7.1) clearly holds. It remains to show the second inclusion, i.e. that if σ∈H and ε∈E⁢U2⁢n+1⁢(R,Δ), then [σ,ε]∈U2⁢n+1⁢((R,Δ),(I,Ω)). Since U2⁢n+1⁢((R,Δ),(I,Ω)) is normalized by E⁢U2⁢n+1⁢(R,Δ) (see part (b) of Remark 6.28), we can assume that ε is an elementary matrix. By the previous theorem and Lemma 6.30, we have σ∈C⁢U2⁢n+1⁢((R,Δ),(I,ΩmaxI)) and therefore [σ,ε]∈U2⁢n+1⁢((R,Δ),(I,ΩmaxI)). Hence, by Lemma 6.25 it remains to show that q⁢([σ,ε]*j)∈Ω for all j∈Θh⁢b and (-.⁡q⁢([σ,ε]*0)(1,0))∘a∈Ω for all a∈J⁢(Δ). But applying the previous theorem to [σ,ε], we get that q⁢([σ,ε]*j)∈Ω for all j∈Θh⁢b. That (-.⁡q⁢([σ,ε]*0)(1,0))∘a∈Ω for all a∈J⁢(Δ) follows from the previous theorem and [2, Lemma 63].

Suppose now that (7.1) holds for some odd form ideal (I,Ω). Then it follows from the standard commutator formula in Theorem 6.31 that H is normalized by E⁢U2⁢n+1⁢(R,Δ). ∎

Communicated by John S. Wilson

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Received: 2018-01-02

Published Online: 2018-04-17

Published in Print: 2018-07-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2018-0011

Sandwich classification for O2n+1(R) and U2n+1(R,Δ) revisited

Article

Abstract

1 Introduction

SCT.

2 Notation

3 Preliminaries

Lemma 3.1.

Lemma 3.2.

4 Odd-dimensional orthogonal groups

4.1 The odd-dimensional orthogonal group

Definition 4.1.

Remark 4.2.

Definition 4.3.

Lemma 4.4.

Proof.

4.2 The polarity map

Definition 4.5.

Lemma 4.6.

Proof.

4.3 The elementary subgroup

Definition 4.7.

Lemma 4.8.

Proof.

Definition 4.9.

Definition 4.10.

Lemma 4.11.

Proof.

4.4 Relative elementary subgroups

Definition 4.12.

Definition 4.13.

4.5 Congruence subgroups

Definition 4.14.

Theorem 4.15.

Proof.

Definition 4.16.

Lemma 4.17.

Proof.

Theorem 4.18.

Proof.

5 Sandwich classification for O2⁢n+1⁢(R)

Definition 5.1.

Theorem 5.2.

Proof.

Corollary 5.3.

Proof.

6 Odd-dimensional unitary groups

6.1 Hermitian form rings and odd form ideals

Definition 6.1.

Remark 6.2.

Definition 6.3.

Definition 6.4.

Remark 6.5.

Definition 6.6.

Definition 6.7.

Lemma 6.8.

6.2 The odd-dimensional unitary group

Lemma 6.9.

Proof.

Definition 6.10.

Remark 6.11.

Example 6.12.

Definition 6.13.

Lemma 6.14.

Proof.

6.3 The polarity map

Definition 6.15.

Lemma 6.16.

Proof.

6.4 The elementary subgroup

Definition 6.17.

Lemma 6.18.

Proof.

Definition 6.19.

Definition 6.20.

Lemma 6.21.

Proof.

Lemma 6.22.

Proof.

6.5 Relative elementary subgroups

Sandwich classification for O_2n+1(R) and U_2n+1(R,Δ) revisited