Commutators of relative and unrelative elementary unitary groups
aa r X i v : . [ m a t h . R A ] A p r COMMUTATORS OF RELATIVE AND UNRELATIVEELEMENTARY UNITARY GROUPS
N. VAVILOV AND Z. ZHANG
Abstract.
In the present paper, which is an outgrowth of our joint work with An-thony Bak and Roozbeh Hazrat on unitary commutator calculus [9, 27, 30, 31], wefind generators of the mixed commutator subgroups of relative elementary groupsand obtain unrelativised versions of commutator formulas in the setting of Bak’sunitary groups. It is a direct sequel of our papers [71, 76, 78, 79] and [77, 80],where similar results were obtained for GL( n, R ) and for Chevalley groups over acommutative ring with 1, respectively. Namely, let ( A, Λ) be any form ring and n ≥
3. We consider Bak’s hyperbolic unitary group GU(2 n, A,
Λ). Further, let( I, Γ) be a form ideal of ( A, Λ). One can associate with ( I, Γ) the corresponding el-ementary subgroup FU(2 n, I,
Γ) and the relative elementary subgroup EU(2 n, I,
Γ)of GU(2 n, A,
Λ). Let ( J, ∆) be another form ideal of ( A, Λ). In the present pa-per we prove an unexpected result that the non-obvious type of generators for[ EU(2 n, I, Γ) , EU(2 n, J, ∆)], as constructed in our previous papers with Hazrat, areredundant and can be expressed as products of the obvious generators, the elemen-tary conjugates Z ij ( ab, c ) = T ji ( c ) T ij ( ab ) T ji ( − c ) and Z ij ( ba, c ), and the elementarycommutators Y ij ( a, b ) = [ T ji ( a ) , T ij ( b )], where a ∈ ( I, Γ), b ∈ ( J, ∆), c ∈ ( A, Λ). Itfollows that [ FU(2 n, I, Γ) , FU(2 n, J, ∆)] = [ EU(2 n, I, Γ) , EU(2 n, J, ∆)]. In fact, weestablish much more precise generation results. In particular, even the elementarycommutators Y ij ( a, b ) should be taken for one long root position and one short rootposition. Moreover, Y ij ( a, b ) are central modulo EU(2 n, ( I, Γ) ◦ ( J, ∆)) and behave assymbols. This allows us to generalise and unify many previous results,including themultiple elementary commutator formula, and dramatically simplify their proofs. To our dear friend Mohammad Reza Darafsheh,with affection and admiration
Introduction
In a series of our joint papers with Anthony Bak and Roozbeh Hazrat [9, 27, 30, 31]we studied commutator formulas in Bak’s unitary groups. In the present paper wegeneralise, refine and strengthen some of the main results of these works. Namely,we discover that the set of generators for the mixed commutator subgroup of relative
Key words and phrases.
Bak’s unitary groups, elementary subgroups, congruence subgroups,standard commutator formula, unrelativised commutator formula, elementary generators, multiplecommutator formula.The work of the first author was supported by the Russian Science Foundation grant 17-11-01261. elementary unitary groups listed in these papers can be substantially reduced andremove all commutativity conditions therein . This allows us to prove unexpectedunrelative versions of the commutator formulas, generalise multiple elementary com-mutator formulas, and more. These results both improve a great number of previousresults, and path the way to several new unexpected applications.Morally, the present paper is a direct sequel our papers [71, 76, 78, 79] and [77, 80],where the same was done for GL( n, R ) and for Chevalley groups over a commutativering with 1, respectively. There, the proofs heavily relied on our previous works, inparticular on [65, 74, 75, 32, 33] for GL( n, R ) and on [28, 29] for Chevalley groups.Similarly, the present paper heavily hinges on the results of [9, 27, 30, 31].0.1. The prior state of art.
To enunciate the main results of the present papers,let us briefly recall the notation, which will be reviewed in somewhat more detail in §§ A, Λ) be a form ring, n ≥
3, and let GU(2 n, A,
Λ) be the hyperbolicBaks unitary group. Below, EU(2 n, A,
Λ) denotes the [absolute] elementary unitarygroup, generated by the elementary root unipotents.As usual, for a form ideal ( I, Γ) of the form ring ( A, Λ) we denote by FU(2 n, I,
Γ)the unrelative elementary subgroup of level ( I, Γ), and by EU(2 n, I,
Γ) the relativeelementary subgroup of level ( I, Γ). By definition, EU(2 n, I,
Γ) is the normal clo-sure of FU(2 n, I,
Γ) in EU(2 n, A,
Λ). Further, GU(2 n, I,
Γ) and CU(2 n, I,
Γ) denotethe principal congruence subgroup and the full congruence subgroup of level ( I, Γ),respectively.Let us recapitulate two principal results of our joint papers with Roozbeh Hazrat,[27, 30, 31]. The first one is the birelative standard commutator formula, [27], Theo-rems 1 and 2. It is a very broad generalisation of the commutator formulas for unitarygroups, previously established by Anthony Bak, the first author, Leonid Vaserstein,Hong You, Gnter Habdank, and others, see, for instance [1, 2, 9, 69, 17, 18, 6].
Theorem A.
Let R be a commutative ring, ( A, Λ) be a form ring such that A is aquasi-finite R -algebra. Further, let ( I, Γ) and ( J, ∆) be two form ideals of the formring ( A, Λ) and let n ≥ . Then the following commutator identity holds [GU(2 n, I, Γ) , EU(2 n, J, ∆)] = [EU(2 n, I, Γ) , EU(2 n, J, ∆)] . When A is itself commutative, one even has [CU(2 n, I, Γ) , EU(2 n, J, ∆)] = [EU(2 n, I, Γ) , EU(2 n, J, ∆)] . Another crucial result is description of a generating set for the mixed commutatorsubgroup [EU(2 n, I, Γ) , EU(2 n, J, ∆)] as a group , similar to the familiar generatingset for relative elementary subgroups, see [9], Proposition 5.1 (compare Lemma 3below).Recall that we denote by T ij ( a ) elementary unitary transvections. They come intwo denominations, those of short root type , when i = ± j , and those of long root In particular, this solves [23], Problem 1 and [30], Problem 1.
OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 3 type , when i = − j . The corresponding root subgroups are then parametrised by thering A itself and by the form parameter Λ, respectively. To simplify notation in therelative case, we introduce the following convention. For a form ideal ( I, Γ) we write a ∈ ( I, Γ) to denote that a ∈ I if i = ± j , and a ∈ λ − ( ε ( i )+1) / Γ if i = − j . Clearly, a ∈ ( I, Γ) means precisely that T ij ( a ) ∈ EU(2 n, I,
Γ), see §§ Z ij ( a, c ) and the elementary com-mutators Y ij ( a, b ), which are defined as follows: Z ij ( a, c ) = T ji ( c ) T ij ( a ) T ji ( − c ) , Y ij ( a, b ) = [ T ji ( a ) , T ij ( b )] , The following result in a slightly weaker form was stated as Theorem 9 of [31], andin precisely this form as Theorem 3B of [30]. Observe that there its proof dependedon Theorem A, and thus ultimately, on localisation methods.
Theorem B.
Let R be a commutative ring, ( A, Λ) be a form ring such that A isa quasi-finite R -algebra. Let ( I, Γ) and ( J, ∆) be two form ideals of the form ring ( A, Λ) and let n ≥ . The relative commutator subgroup [EU(2 n, I, Γ) , EU(2 n, J, ∆)] is generated by the elements of the following three types • Z ij ( ab, c ) and Z ij ( ba, c ) , • Y ij ( a, b ) , • [ T ij ( a ) , Z ij ( b, c )] ,where in all cases a ∈ ( I, Γ) , b ∈ ( J, ∆) and c ∈ ( A, Λ)0.2.
Statement of the principal result.
The technical core of the present paperare Lemmas 6–12 that we prove in §§ drastically generalised and improved, as follows: • We can lift the commutativity condition. • The third type of generators are redundant. • The second type of generators can be restricted to one long and one short root(and are subject to further relations, to be stated below).The following result is the pinnacle of the present paper, other results are eitherpreparation to its proof, or its easy corollaries. For the general linear group GL( n, R )it was established in [76], Theorem 1. For Chevalley groups G (Φ , R ) over commutativerings — and thus, in particular, for the usual symplectic group Sp(2 n, R ) and thesplit orthogonal group SO(2 n, R ) — it is essentially a conjunction of [77], Theorem1.2, and [80], Theorem 1. However, as explained below, in these special cases one cansay somewhat more. Theorem 1.
Let ( A, Λ) be any associative form ring, let ( I, Γ) and ( J, ∆) be two formideals of the form ring ( A, Λ) and let n ≥ . Then the relative commutator subgroup [EU(2 n, I, Γ) , EU(2 n, J, ∆)] is generated by the elements of the following two types • Z ij ( ab, c ) and Z ij ( ba, c ) , N. VAVILOV AND Z. ZHANG • Y ij ( a, b ) ,where in all cases a ∈ ( I, Γ) , b ∈ ( J, ∆) and c ∈ ( A, Λ) . Moreover, for the secondtype of generators it suffices to take one pair ( h, k ) , h = ± k , and one pair ( h, − h ) . The difference with Chevalley groups is that now we have to throw in elementarycommutators for two roots, one long root and one short root. For Chevalley groups,one long root would suffice. Conversely, when 2 is invertible for types B l , C l , F and3 is invertible for type G , one short toot would suffice. For unitary groups, moduloEU(2 n, ( I, Γ) ◦ ( J, ∆)) we can still establish a cognate relation between short root typeelementary commutators and long root type elementary commutators, Lemma 12.However, unlike Chevalley groups, for unitary groups the elements of long root sub-groups are parametrised by the form parameter Λ, whereas the elements of shortroot subgroups are parametrised by the ring A itself. This means that now we coulddispose of some short type elementary commutators, yet not all of them. In theopposite direction, the long type elementary commutators, one of whose argumentssits in the corresponding minimal ideal form parameter could be discarded — butnot all of them! This can be done when one of the form parameters is either minimal,or as large as possible — not just maximal! — see § n, I, Γ) , EU(2 n, J, ∆)] will be called the firstclaim of Theorem 1. The much more arduous bid that modulo EU(2 n, ( I, Γ) ◦ ( J, ∆))all elementary commutators can be expressed in terms of such commutators in oneshort and one long positions, will be called the second claim of Theorem 1.Let us mention another important trait. The published proofs of Theorem B heavilydepended on some version of Theorem A, and thus, ultimately, on localisation. Theproof of Theorem 1 given below in §§ elementary and thus works alreadyat the level of unitary Steinberg groups , see [1, 2, 36]. The only reason why we do notstate our results in this generality is to skip discussion of relative unitary Steinberggroups . The details and technical facts are not readily available in the literature, andwould noticeably increase the length of the present paper.0.3. Unrelativisation.
Since both remaining types of generators listed in Theorem 1already belong to the mixed commutator of the unrelative elementary subgroups[FU(2 n, I, Γ) , FU(2 n, J, ∆)], we get the following amazing equality. Morally, it showsthat the commutator of relative elementary subgroups [EU(2 n, I, Γ) , EU(2 n, J, ∆)]is smaller, than one expects. Observe that it only depends on the [relatively] easyfirst claim of Theorem 1 whose proof is completed already in §
5. For GL( n, R ) thecorresponding result is [71], Theorem 2 (for commutative rings, with a completelydifferent proof), and [76], Theorem 1 (for arbitrary associative rings). For Sp(2 n, R )and SO(2 n, R ) it is a special case of [77], Theorem 1.2. In the technical sense that it does not invoke anything apart from the usual Steinberg relations.
OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 5
Theorem 2.
Let ( A, Λ) be any associative form ring, let ( I, Γ) and ( J, ∆) be two formideals of the form ring ( A, Λ) and let n ≥ . Then the mixed commutator subgroup [FU(2 n, I, Γ) , FU(2 n, J, ∆)] is normal in
EU(2 n, A, Λ) . Furthermore, we have thefollowing commutator identity [FU(2 n, I, Γ) , FU(2 n, J, ∆)] = [EU(2 n, I, Γ) , EU(2 n, J, ∆)] . In particular, in conjunction with Theorem A this shows that the birelative stan-dard commutator formula also holds in the following unrelativised form. Again, forGL( n, R ) this is [71], Theorem 1 and [76], Theorem 3, whereas for Chevalley groupsit is [77], Theorem 1.3.
Theorem 3.
Let R be a commutative ring, ( A, Λ) be a form ring such that A is aquasi-finite R -algebra. Further, let ( I, Γ) and ( J, ∆) be two form ideals of the formring ( A, Λ) and let n ≥ . Then we have a unrelative commutator identity [GU(2 n, I, Γ) , EU(2 n, J, ∆)] = [FU(2 n, I, Γ) , FU(2 n, J, ∆)] . When A is itself commutative, one even has [CU(2 n, I, Γ) , EU(2 n, J, ∆)] = [FU(2 n, I, Γ) , FU(2 n, J, ∆)] . The following result is a unitary analogue of the unrelative normality theoremproven for GL( n, R ) by Bogdan Nica and ourselves, see [44, 71, 78]. It is an immediatecorollary of our Theorem 3, if you set there ( I, Γ) = ( J, ∆). Theorem 4.
Let R be a commutative ring, ( A, Λ) be a form ring such that A is aquasi-finite R -algebra. Further, let ( I, Γ) be a form ideals of the form ring ( A, Λ) andlet n ≥ . Then FU(2 n, I, Γ) is normal in GU( n, I, Γ) . Elementary commutators.
The proof of the second claim of Theorem 1 isthe gist of the present paper, and proceeds as follows. First, in § Y ij ( a, b ) are central in the absolute elementary groupmodulo EU(2 n, ( I, Γ) ◦ ( J, ∆)). Recall that here( I, Γ) ◦ ( J, ∆) = ( IJ + J I, J Γ + I ∆ + Γ min ( IJ + J I ))denotes the symmetrised product of form ideals, see § n, ( I, Γ) ◦ ( J, ∆))these commutators generate [FU(2 n, I, Γ) , FU(2 n, J, ∆)], this result can be statedas follows. For GL( n, R ) and Chevalley groups this is [76], Theorem 2, and [80],Theorem 2, respectively.
Theorem 5.
Let ( A, Λ) be any associative form ring, let ( I, Γ) and ( J, ∆) be twoform ideals of the form ring ( A, Λ) and let n ≥ . Then [FU(2 n, I, Γ) , FU(2 n, J, ∆)] is central in
EU(2 n, A, Λ) modulo EU(2 n, ( I, Γ) ◦ ( J, ∆)) . In other words, (cid:2) [ FU(2 n, I, Γ) , FU(2 n, J, ∆)] , EU(2 n, A, Λ) (cid:3) ≤ EU(2 n, ( I, Γ) ◦ ( J, ∆)) . N. VAVILOV AND Z. ZHANG
In particular, it implies that the quotient[ FU(2 n, I, Γ) , FU(2 n, J, ∆)] / EU(2 n, ( I, Γ) ◦ ( J, ∆))is itself abelian. This readily implies additivity of the elementary commutator withrespect to its arguments, and other similar useful properties, collected in Theorem 10,that are employed in the proofs of subsequent results.However, the focal point of the present paper is §
7, where we prove that moduloEU(2 n, ( I, Γ) ◦ ( J, ∆)) all elementary commutators of the same root type are equiva-lent. Moreover, for the short root type they are balanced with respect to the factorsfrom R , both on the left and on the right. For the long root type the balancing prop-erty is more complicated, and only holds for the quadratic (=Jordan) multiplication.In the case of the usual symplectic group, where A is a commutative ring with trivialinvolution, it corresponds to the multiplication by squares, see [80], Theorem 5. Theorem 6.
Let ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) , ( J, ∆) be form ideals of ( A, Λ) . • Then for any i = ± j , any h = ± l with h, l = ± i, ± j , and a ∈ I , b ∈ J , c, d ∈ A ,the elementary commutator Y ij ( cad, b ) ≡ Y hl ( a, dbc ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . • Then for any − n ≤ i ≤ n , any − n ≤ k ≤ n , and a ∈ λ − ( ε ( i )+1) / Γ , b ∈ λ ( ε ( i ) − / ∆ c ∈ A , the elementary commutator Y i, − i ( cac, b ) ≡ Y k, − k ( λ ( ε ( i ) − ε ( k )) / a, − λ ( ε ( k ) − ε ( i )) / cbc ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . The calculation behind these congruences is the highlight of the whole theory.Inherently, it is just a birelative incarnation of a classical calculation that appeareddozens of times in the algebraic K-theory and the theory of algebraic groups sincemid 60-ies, see §
12 for a terse historical medley.0.5.
Further corollaries.
As another illustration of the power of Theorem 1, weshow that it allows to [almost completely] lift commutativity conditions in some ofthe principal results of [27, 30, 31].Under the additional assumptions such as quasi-finiteness the following result forany n ≥ n ≥ n, R ) such generalisation wasalready obtained in [76]. We believe this could be also done for n = 3, see Problem3, but in that case it would require formidable calculations. Theorem 7.
Let ( A, Λ ) be any associative form ring with , let n ≥ , and let ( I i , Γ i ) E R , i = 1 , . . . , m , be form ideals of ( A, Λ) . Consider an arbitrary arrange-ment of brackets J . . . K with the cut point s . Then one has q EU(2 n, I , Γ ) , EU(2 n, I , Γ ) , . . . , EU(2 n, I m , Γ m ) y = (cid:2) EU(2 n, ( I , Γ ) ◦ . . . ◦ ( I s , Γ s )) , EU(2 n, ( I s +1 , Γ s +1 ) ◦ . . . ◦ ( I m , Γ m ) (cid:3) , OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 7 where the bracketing of symmetrised products on the right hand side coincides withthe bracketing of the commutators on the left hand side.
Under the additional assumption that the absolute standard commutator formulaeare satisfied, the following result is [27], Theorem 3. As we know from [9, 20, 21, 27],this condition is satisfied for quasi-finite rings. But from the work of Victor Gerasimov[16] it follows that some commutativity or finiteness assumptions are necessary forthe standard commutator formulae to hold. Now, we are in a position to prove thefollowing result for arbitrary associative form rings.
Theorem 8.
Let ( A, Λ) be any associative form ring and n ≥ . Then for any twocomaximal form ideals ( I, Γ) and ( J, ∆) of the form ring ( R, Λ) , I + J = A , one hasthe following equality [EU(2 n, I, Γ) , EU(2 n, J, ∆)] = EU(2 n, ( I, Γ) ◦ ( J, ∆)) . Another bizarre corollary of Theorem 1 is surjective stability of the quotients[FU(2 n, I, Γ) , FU(2 n, J, ∆)] / EU(2 n, ( I, Γ) ◦ ( J, ∆)) , again for arbitrary associative form rings, without any stability conditions, or com-mutativity conditions. This is a typical result in the style of Bak’s paradigm “stabilityresults without stability conditions”, see [3] and also [20, 21, 25, 26, 4]. Theorem 9.
Let ( A, Λ) be any associative form ring, let ( I, Γ) and ( J, ∆) be twoform ideals of the form ring ( A, Λ) and let n ≥ . Then the stability map [FU(2 n, I, Γ) , FU(2 n, J, ∆)] / EU(2 n, ( I, Γ) ◦ ( J, ∆)) −→ [FU(2( n + 1) , I, Γ) , FU(2( n + 1) , J, ∆)] / EU(2( n + 1) , ( I, Γ) ◦ ( J, ∆)) is surjective. Indeed, in view of Theorems 1 and 5 as a normal subgroup of EU(2 n, A,
Λ) thegroup [EU(2 n, I, Γ) , EU(2 n, J, ∆)] is generated by [EU(6 , I, Γ) , EU(6 , J, ∆)]. An ex-plicit calculation of these quotients presents itself as a natural next step. However,so far we were unable to resolve it, apart from some special cases, see a discussion in § Organisation of the paper.
The rest of the paper is devoted to the proofof these results. In §§ §§ § § n, I, Γ) , EU(2 n, J, ∆)] to the first two types. In § § some elementary commutators of short root type with some elementary commutatorsof long root type. This finishes the proof of Theorem 1 and its corollaries, and, inparticular, also of Theorems 2–4 In § §
10 derive Theorem 7itself by an easy induction. In §
11 we derive Theorem 8 and yet another corollary
N. VAVILOV AND Z. ZHANG of our main results. Finally, in §
12 we describe the general context, briefly reviewrecent related publications and state several further related open problems.1.
Notation
Here we recall some basic notation that will be used throughout the present paper.1.1.
General linear group.
Let, as above, A be an associative ring with 1. Fornatural m, n we denote by M ( m, n, A ) the additive group of m × n matrices withentries in A . In particular M ( m, A ) = M ( m, m, A ) is the ring of matrices of degree m over A . For a matrix x ∈ M ( m, n, A ) we denote by x ij , 1 ≤ i ≤ m , 1 ≤ j ≤ n ,its entry in the position ( i, j ). Let e be the identity matrix and e ij , 1 ≤ i, j ≤ m , bea standard matrix unit, i.e. the matrix which has 1 in the position ( i, j ) and zeroselsewhere.As usual, GL( m, A ) = M ( m, A ) ∗ denotes the general linear group of degree m over A . The group GL( m, A ) acts on the free right A -module V ∼ = A m of rank m . Fix abase e , . . . , e m of the module V . We may think of elements v ∈ V as columns withcomponents in A . In particular, e i is the column whose i -th coordinate is 1, while allother coordinates are zeros.Actually, in the present paper we are only interested in the case, when m = 2 n iseven. We usually number the base as follows: e , . . . , e n , e − n , . . . , e − . All otheroccurring geometric objects will be numbered accordingly. Thus, we write v =( v , . . . , v n , v − n , . . . , v − ) t , where v i ∈ A , for vectors in V ∼ = A n .The set of indices will be always ordered in conformity with this convention, Ω = { , . . . , n, − n, . . . , − } . Clearly, Ω = Ω + ⊔ Ω − , where Ω + = { , . . . , n } and Ω − = {− n, . . . , − } . For an element i ∈ Ω we denote by ε ( i ) the sign of Ω, i.e. ε ( i ) = +1if i ∈ Ω + , and ε ( i ) = − i ∈ Ω − .1.2. Commutators.
Let G be a group. For any x, y ∈ G , x y = xyx − and y x = x − yx denote the left conjugate and the right conjugate of y by x , respectively. Asusual, [ x, y ] = xyx − y − denotes the left-normed commutator of x and y . Throughoutthe present paper we repeatedly use the following commutator identities:(C1) [ x, yz ] = [ x, y ] · y [ x, z ],(C1 + ) An easy induction, using identity (C1), shows that (cid:20) x, k Y i =1 u i (cid:21) = k Y i =1 Q i − j =1 u j [ x, u i ] , where by convention Q j =1 u j = 1,(C2) [ xy, z ] = x [ y, z ] · [ x, z ],(C2 + ) As in (C1 + ), we have (cid:20) k Y i =1 u i , x (cid:21) = k Y i =1 Q k − ij =1 u j [ u k − i +1 , x ] , OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 9 (C3) x [[ x − , y ] , z ] · z [[ z − , x ] , y ] · y [[ y − , z ] , x ] = 1,(C4) [ x, y z ] = y [ y − x, z ],(C5) [ y x, z ] = y [ x, y − z ],(C6) If H and K are subgroups of G , then [ H, K ] = [
K, H ],Especially important is (C3), the celebrated
Hall–Witt identity . Sometimes it is usedin the following form, known as the three subgroup lemma . Lemma 1.
Let
F, H, L E G be three normal subgroups of G . Then [[ F, H ] , L ] ≤ [[ F, L ] , H ] · [ F, [ H, L ]] . Form rings and form ideal
The notion of Λ-quadratic forms, quadratic modules and generalised unitary groupsover a form ring ( A, Λ) were introduced by Anthony Bak in his Thesis, see [1, 2]. Inthis section, and the next one, we very briefly review the most fundamental notationand results that will be constantly used in the sequel. We refer to [11, 2, 19, 35, 9,67, 20, 21, 46, 27, 30, 31, 36] for details, proofs, and further references. In the finalsection we mention some further related recent works, and some generalisations.2.1.
Form rings.
Let R be a commutative ring with 1, and A be an (not necessarilycommutative) R -algebra. An involution, denoted by , is an anti-homomorphism of A of order 2. Namely, for a, b ∈ A , one has a + b = a + b, ab = b a, a = a. Fix an element λ ∈ Cent( A ) such that λλ = 1. One may define two additive sub-groups of A as follows:Λ min = { c − λc | c ∈ A } , Λ max = { c ∈ A | c = − λc } . A form parameter Λ is an additive subgroup of A such that(1) Λ min ⊆ Λ ⊆ Λ max ,(2) c Λ c ⊆ Λ for all c ∈ A .The pair ( A, Λ) is called a form ring .2.2.
Form ideals.
Let I E A be a two-sided ideal of A . We assume I to be involutioninvariant, i. e. such that I = I . SetΓ max ( I ) = I ∩ Λ , Γ min ( I ) = { a − λa | a ∈ I } + h aca | a ∈ I, c ∈ Λ i . A relative form parameter Γ in ( A, Λ) of level I is an additive group of I such that(1) Γ min ( I ) ⊆ Γ ⊆ Γ max ( I ),(2) c Γ c ⊆ Γ for all c ∈ A . The pair ( I, Γ) is called a form ideal .In the level calculations we will use sums and products of form ideals. Let ( I, Γ)and ( J, ∆) be two form ideals. Their sum is artlessly defined as ( I + J, Γ + ∆), it isimmediate to verify that this is indeed a form ideal.Guided by analogy, one is tempted to set ( I, Γ)( J, ∆) = ( IJ,
Γ∆). However, it isconsiderably harder to correctly define the product of two relative form parameters.The papers [17, 18, 20, 21] introduce the following definitionΓ∆ = Γ min ( IJ ) + J Γ + I ∆ , where J Γ = h b Γ b | b ∈ J i , I ∆ = h a ∆ a | a ∈ I i . One can verify that this is indeed a relative form parameter of level IJ if IJ = J I .However, in the present paper we do not wish to impose any such commutativityassumptions. Thus, we are forced to consider the symmetrised products I ◦ J = IJ + J I, Γ ◦ ∆ = Γ min ( IJ + J I ) + J Γ + I ∆The notation Γ ◦ ∆ – as also Γ∆ is slightly misleading, since in fact it depends on I and J , not just on Γ and ∆. Thus, strictly speaking, one should speak of thesymmetrised products of form ideals ( I, Γ) ◦ ( J, ∆) = ( IJ + J I, Γ min ( IJ + J I ) + J Γ + I ∆) . Clearly, in the above notation one has( I, Γ) ◦ ( J, ∆) = ( I, Γ)( J, ∆) + ( J, ∆)( I, Γ) . Unitary groups
In the present section we recall basic notation and facts related to Bak’s generalisedunitary groups.3.1.
Unitary group.
For a form ring ( A, Λ), one considers the hyperbolic unitarygroup
GU(2 n, A,
Λ), see [9, § λ ∈ Cent( A ), λλ = 1 and supplies the module V = A n withthe following λ -hermitian form h : V × V −→ A , h ( u, v ) = u v − + . . . + u n v − n + λu − n v n + . . . + λu − v . and the following Λ-quadratic form q : V −→ A/ Λ, q ( u ) = u u − + . . . + u n u − n mod Λ . In fact, both forms are engendered by a sesquilinear form f , f ( u, v ) = u v − + . . . + u n v − n . Now, h = f + λf , where f ( u, v ) = f ( v, u ), and q ( v ) = f ( u, u ) mod Λ. OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 11
By definition, the hyperbolic unitary group GU(2 n, A,
Λ) consists of all elementsfrom GL( V ) ∼ = GL(2 n, A ) preserving the λ -hermitian form h and the Λ-quadraticform q . In other words, g ∈ GL(2 n, A ) belongs to GU(2 n, A,
Λ) if and only if h ( gu, gv ) = h ( u, v ) and q ( gu ) = q ( u ) , for all u, v ∈ V. When the form parameter is neither maximal nor minimal, these groups are notalgebraic. However, their internal structure is very similar to that of the usual classicalgroups. They are also oftentimes called general quadratic groups, or classical-likegroups.3.2.
Unitary transvections.
Elementary unitary transvections T ij ( ξ ) correspondto the pairs i, j ∈ Ω such that i = j . They come in two stocks. Namely, if, moreover, i = − j , then for any c ∈ A we set T ij ( c ) = e + ce ij − λ ( ε ( j ) − ε ( i )) / ce − j, − i . These elements are also often called elementary short root unipotents . On the otherside for j = − i and c ∈ λ − ( ε ( i )+1) / Λ we set T i, − i ( c ) = e + ce i, − i . These elements are also often called elementary long root elements .Note that Λ = λ Λ. In fact, for any element c ∈ Λ one has c = − λc and thus Λcoincides with the set of products λc , where c ∈ Λ. This means that in the abovedefinition c ∈ Λ when i ∈ Ω + and c ∈ Λ when i ∈ Ω − .Subgroups X ij = { T ij ( c ) | c ∈ A } , where i = ± j , are called short root subgroups .Clearly, X ij = X − j, − i . Similarly, subgroups X i, − i = { T ij ( c ) | c ∈ λ − ( ε ( i )+1) / Λ } arecalled long root subgroups .The elementary unitary group EU(2 n, A,
Λ) is generated by elementary unitarytransvections T ij ( c ), i = ± j , c ∈ A , and T i, − i ( c ), c ∈ Λ, see [9, § Steinberg relations.
Elementary unitary transvections T ij ( ξ ) satisfy the fol-lowing elementary relations , also known as Steinberg relations . These relations willbe used throughout this paper.(R1) T ij ( c ) = T − j, − i ( − λ ( ε ( j ) − ε ( i )) / c ),(R2) T ij ( c ) T ij ( d ) = T ij ( c + d ),(R3) [ T ij ( c ) , T hk ( d )] = e , where h = j, − i and k = i, − j ,(R4) [ T ij ( c ) , T jh ( d )] = T ih ( cd ), where i, h = ± j and i = ± h ,(R5) [ T ij ( c ) , T j, − i ( d )] = T i, − i ( cd − λ − ε ( i ) dc ), where i = ± j ,(R6) [ T i, − i ( a ) , T − i,j ( d )] = T ij ( ac ) T − j,j ( − λ ( ε ( j ) − ε ( i )) / cac ), where i = ± j .Relation (R1) coordinates two natural parametrisations of the same short root sub-group X ij = X − j, − i . Relation (R2) expresses additivity of the natural parametrisa-tions. All other relations are various instances of the Chevalley commutator formula. Namely, (R3) corresponds to the case, where the sum of two roots is not a root,whereas (R4), and (R5) correspond to the case of two short roots, whose sum is ashort root, and a long root, respectively. Finally, (R6) is the Chevalley commutatorformula for the case of a long root and a short root, whose sum is a root. Observethat any two long roots are either opposite, or orthogonal, so that their sum is nevera root. 4.
Relative subgroups
In this section we recall definitions and basic facts concerning relative subgroups.For the proofs of these results, see4.1.
Relative subgroups.
One associates with a form ideal ( I, Γ) the following fourrelative subgroups. • The subgroup FU(2 n, I,
Γ) generated by elementary unitary transvections of level( I, Γ),FU(2 n, I,
Γ) = h T ij ( a ) | a ∈ I if i = ± j and a ∈ λ − ( ε ( i )+1) / Γ if i = − j i . • The relative elementary subgroup
EU(2 n, I,
Γ) of level ( I, Γ), defined as the nor-mal closure of FU(2 n, I,
Γ) in EU(2 n, A,
Λ),EU(2 n, I,
Γ) = FU(2 n, I, Γ) EU(2 n,A, Λ) . • The principal congruence subgroup
GU(2 n, I,
Γ) of level ( I, Γ) in GU(2 n, A,
Λ)consists of those g ∈ GU(2 n, A,
Λ), which are congruent to e modulo I and preserve f ( u, u ) modulo Γ, f ( gu, gu ) ∈ f ( u, u ) + Γ , u ∈ V. • The full congruence subgroup CU(2 n, I,
Γ) of level ( I, Γ), defined asCU(2 n, I,
Γ) = { g ∈ GU(2 n, A, Λ) | [ g, GU(2 n, A,
Λ)] ⊆ GU(2 n, I, Γ) } . In some books, including [19], the group CU(2 n, I,
Γ) is defined differently. How-ever, in many important situations these definitions yield the same group.4.2.
Some basic lemmas.
Let us collect several basic facts, concerning relativegroups, which will be used in the sequel. The first one of them, see [9], Lemma 5.2,asserts that the relative elementary groups are EU(2 n, A,
Λ)-perfect.
Lemma 2.
Suppose either n ≥ or n = 2 and I = Λ I + I Λ . Then EU(2 n, I,
Γ) = [EU(2 n, I, Γ) , EU(2 n, A,
Λ)] . The next lemma gives generators of the relative elementary subgroup EU(2 n, I,
Γ)as a subgroup. With this end, consider matrices Z ij ( a, c ) = T ji ( c ) T ij ( a ) = T ji ( c ) T ij ( a ) T ji ( − c ) , OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 13 where a ∈ I , c ∈ A , if i = ± j , and a ∈ λ − ( ε ( i )+1) / Γ, c ∈ λ − ( ε ( j )+1) / Λ, if i = − j . Thefollowing result is [9], Proposition 5.1. Lemma 3.
Suppose n ≥ . Then EU(2 n, I,
Γ) = h Z ij ( a, c ) | a ∈ I, c ∈ A if i = ± j and a ∈ λ − ( ε ( i )+1) / Γ , c ∈ λ − ( ε ( j )+1) / Λ , if i = − j i . The following lemma was first established in [1], but remained unpublished. See [19]and [9], Lemma 4.4, for published proofs.
Lemma 4.
The groups
GU(2 n, I, Γ) and CU(2 n, I, Γ) are normal in GU(2 n, A, Λ) . In this form the following lemma was established in [31], Lemmas 7 and 8, see also[30], Lemma 1B for a definitive exposition. Before that [27], Lemmas 21–23 onlyestablished weaker inclusions, with smaller left hand sides, or larger right hand sides.
Lemma 5. ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) and ( J, ∆) be two form ideals of ( A, Λ) . Then EU(2 n, ( I, Γ) ◦ ( J, ∆)) ≤ [ FU(2 n, I, Γ) , FU(2 n, J, ∆)] ≤ [ EU(2 n, I, Γ) , EU(2 n, J, ∆)] ≤ [ GU(2 n, I, Γ) , GU(2 n, J, ∆)] ≤ GU(2 n, ( I, Γ) ◦ ( J, ∆)) . Elementary commutators modulo
EU(2 n, ( I, Γ) ◦ ( J, ∆))Now we embark on the proof of the second claim of Theorem 1. Our first ma-jor goal is to prove that the commutator [FU(2 n, I, Γ) , FU(2 n, J, ∆)] is central inEU(2 n, A,
Λ), modulo EU(2 n, ( I, Γ) ◦ ( J, ∆)). Namely, here we establish Theorem 5and derive some corollaries thereof. We prove the congruence in Theorem 5 separatelyfor short root positions, and then for long root positions. Lemma 6.
Let ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) , ( J, ∆) be form ideals of ( A, Λ) . For any i = ± j any a ∈ I , b ∈ J and any x ∈ EU(2 n, A, Λ) ,one has x Y ij ( a, b ) ≡ Y ij ( a, b ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Proof.
Consider the elementary conjugate x Y ij ( a, b ). We argue by induction on thelength of x ∈ EU(2 n, A,
Λ) in elementary generators. Let x = yT kl ( c ), where y ∈ EU(2 n, A,
Λ) is shorter than x .We start with the case k = ± l . • If k, l = ± i, ± j , then T kl ( c ) commutes with z = Y ij ( a, b ) and can be discarded. • On the other hand, for any h = ± i, ± j direct computations show that [ T ih ( c ) , z ] = T ih ( − abc − ababc ) T jh ( − babc ) , [ T jh ( c ) , z ] = T ih ( abac ) T jh ( bac ) , [ T hi ( c ) , z ] = T ih ( cab ) T jh ( − caba ) , [ T hj ( c ) , z ] = T ih ( cbab ) T jh ( − cba − cbaba ) , Similarly, one has[ T − i,h ( c ) , z ] = [ T − h,i ( − λ (( ε ( h )+ ε ( i )) / c ) , z ] r = T i, − h ( − λ (( ε ( h )+ ε ( i )) / cab ) T j, − h ( − λ (( ε ( h )+ ε ( i )) / caba ) , [ T − j,h ( c ) , z ] = [ T − h,j ( − λ (( ε ( h )+ ε ( j )) / c ) , z ]= T i, − h ( − λ (( ε ( h )+ ε ( j )) / cbab ) T j, − h ( − λ (( ε ( h )+ ε ( j )) / cba − λ (( ε ( h )+ ε ( j )) / cbaba ) , [ T h, − i ( c ) , z ] = [ T i, − h ( − λ ( − ( ε ( i ) − ε ( h )) / c ) , z ]= T i, − h ( − λ ( − ( ε ( i ) − ε ( h )) / abac ) T j, − h ( − λ ( − ( ε ( i ) − ε ( h )) / bac ) , [ T h, − j ( c ) , z ] = [ T j, − h ( − λ ( − ( ε ( j ) − ε ( h )) / c ) , z ]= T i, − h ( − λ ( − ( ε ( j ) − ε ( h )) / abac ) T j, − h ( − λ ( − ( ε ( j ) − ε ( h )) / bac )All factors on the right hand side belong already to EU(2 n, ( I, Γ) ◦ ( J, ∆)).If ( k, l ) = ( ± i, ± j ) or ( ± j, ± i ), then we take an index h = ± i, ± j and rewrite T kl ( c ) as [ T k,h ( c ) , T h,l (1)] and apply the previous items to get the same congruencemodulo EU(2 n, ( I, Γ) ◦ ( J, ∆)).It remains to consider the case, where k = − l . • if k = ± i, ± j then T k, − k ( c ) commutes with z and can be discarded. • Otherwise, we have[ T i, − i ( c ) , z ] = T i, − i ( c − (1 + ab + abab ) c (1 + ab + abab )) T j, − j ( − λ (( ε ( i ) − ε ( j )) / babcbab ) T i, − j ( λ (( ε ( i ) − ε ( j )) / (1 + ab + abab ) c ( bab )) , [ T j, − j ( c ) , z ] = T j, − j ( c − (1 − ba ) c (1 − ba )) T i, − i ( λ (( ε ( j ) − ε ( i )) / abacaba ) T i, − j ( − abac (1 − ba )) , [ T − i,i ( c ) , z ] =[ T − i,i ( c ) , [ T ij ( a ) , T ji ( b )]]=[ T − i,i ( c ) , [ T − j, − i ( − λ (( ε ( j ) − ε ( i )) / a ) , T − i, − j ( λ (( ε ( i ) − ε ( j )) / b )]] , [ T − j,j ( c ) , z ] =[ T − j,j ( c ) , [ T ij ( a ) , T ji ( b )]]=[ T − j, − j ( c ) , [ T − j, − i ( − λ (( ε ( j ) − ε ( i )) / a ) , T − i, − j ( λ (( ε ( i ) − ε ( j )) / b )]] . The two last cases reduce to the first two. Hence all factors on the right belong toEU(2 n, ( I, Γ) ◦ ( J, ∆)). OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 15
We have shown that for i = ± j , x z ≡ y z (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . (cid:3) Lemma 7.
Let ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) , ( J, ∆) be form ideals of ( A, Λ) . For any a ∈ λ − ( ε ( i )+1) / Γ , b ∈ λ ( ε ( i ) − / ∆ and any x ∈ EU(2 n, A, Λ) , one has x Y i, − i ( a, b ) ≡ Y i, − i ( a, b ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Proof.
Denote Y i, − i ( a, b ) = [ T i, − i ( a ) , T − i,i ( b )] by z . • If ( k, l ) = ( − i, i ), then[ T − i,i ( c ) , z ] = [ T − i,i ( c ) , [ T i, − i ( a ) , T − i,i ( b )]] = [ T − i,i ( c ) , Z − i,i ( b, a )] . The same computation as in Case 2 in Lemma 6 shows that[ T − i,i ( c ) , z ] ∈ EU(2 n, ( I, Γ) ◦ ( J, ∆)) . • If ( k, l ) = ( i, − i ), then[ T i, − i ( c ) , z ] =[ T i, − i ( c ) , [ T i, − i ( a ) , T − i,i ( b )]] =[ T i, − i ( c ) , [ T − i,i ( b ) , T i, − i ( a )] − ] =[ T − i,i ( b ) , T i, − i ( a )] − [[ T − i,i ( b ) , T i, − i ( a )] , T i, − i ( c )][ T − i,i ( b ) , T i, − i ( a )] . Now the inner factor [[ T − i,i ( b ) , T i, − i ( a )] , T i, − i ( c )] falls into the previous case, hencebelongs to EU(2 n, ( I, Γ) ◦ ( J, ∆)). But then the same applies also to its conjugate[ T − i,i ( b ) , T i, − i ( a )] − · (cid:2) [ T − i,i ( b ) , T i, − i ( a )] , T i, − i ( c ) (cid:3) · [ T − i,i ( b ) , T i, − i ( a )] . • If k = i and j = ± k , then[ T i,j ( c ) , z ] = [ T i,j ( c ) , [ T i, − i ( a ) , T − i,i ( b )]] = T i,j ( − ( ab + abab ) c ) T − i,j ( − babc ) · T − j,j ( − λ (( ε ( j ) − ε ( i )) / cbabc − λ ε ( j ) ( cbababc + cbabababc )) . Since a ∈ λ − ( ε ( i )+1) / Γ and b ∈ λ ( ε ( i ) − / ∆, it follows that the right side belongs toEU(2 n, ( I, Γ) ◦ ( J, ∆)). • if k = − i and j = ± k , then[ T − i,j ( c ) , z ] =[ T − i,j ( c ) , [ T i, − i ( a ) , T − i,i ( b )]]=[ T − i,i ( b ) , T i, − i ( a )][ T − i,j ( c ) , [ T − i,i ( b ) , T i, − i ( a )]] − [ T − i,i ( b ) , T i, − i ( a )] − . By the previous case,[ T − i,j ( c ) , [ T − i,i ( b ) , T i, − i ( a )]] ∈ EU(2 n, ( I, Γ) ◦ ( J, ∆)) . As above, normality of EU(2 n, ( I, Γ) ◦ ( J, ∆)) then implies that the whole right sidebelongs to EU(2 n, ( I, Γ) ◦ ( J, ∆)). • Finally, the case l = ± i and k = ± i reduces to the case k = ± i via relation (R1). We have shown that x z ≡ y z (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . By induction we get that x z ≡ z (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . (cid:3) In particular, these results immediately imply the following additivity property ofthe elementary commutators with respect to its arguments.
Theorem 10.
Let R be an associative ring with , n ≥ , and let ( I, Γ) , ( J, ∆) beform ideals of R . Then for any i = j , and any a, a , a ∈ ( I, Γ) , b, b , b ∈ ( J, ∆) onehas Y ij ( a + a , b ) ≡ Y ij ( a , b ) · Y ij ( a , b ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) ,Y ij ( a, b + b ) ≡ Y ij ( a, b ) · Y ij ( a, b ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) ,Y ij ( a, b ) − ≡ Y ij ( − a, b ) ≡ Y ij ( a, − b ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) ,Y ij ( ab , b ) ≡ Y ij ( a , a b ) ≡ e (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) Y i, − i ( b ab , b ) ≡ Y i, − i ( a , a ba ) ≡ e (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) Proof.
The first item can be derived from Lemma 10 for i = ± j and Lemma 11 for i = − j as follows. By definition Y ij ( a + a , b ) = [ T ij ( a + a ) , T ji ( b )] = [ T ij ( a ) T ij ( a ) , T ji ( b )] , and it only remains to apply multiplicativity of commutators in the first factor, andthen apply Lemma 10 and Lemma 11 respectively. The second item is similar, andthe third item follows. The last two items are obvious from the definition. (cid:3) Unrelativisation
Here we establish the first claim of Theorem 1, and thus also Theorems 2, 3 and4. It immediately follows from the next two lemmas, the first of which addresses thecase of short roots, while the second one the case of long roots.Recall that for the easier case of the general linear group over commutative ringsthis result was first established in 2018 in our paper [77]. Then it was generalised toarbitrary associative rings in 2019, together with the second claim of Theorem 1, see[76]. The proof of the following results exploit the same ideas as the proof of [76],Lemma 4, but are noticeably more demanding from a technical viewpoint.The following two lemmas address the case of short roots, where i = ± j , and thecase of long roots, where i = − j , respectively OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 17
Lemma 8.
Let ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) , ( J, ∆) be form ideals of ( A, Λ) . Suppose that a ∈ I , b ∈ J , r ∈ A and i = ± j . Then [ T ji ( a ) , Z ji ( b, r )] ∈ [FU(2 n, I, Γ) , FU(2 n, J, ∆)] . Proof.
Without loss of generality, we may assume that ε ( i ) = ε ( j ). Pick an h = i, j with ε ( h ) = ε ( i ). Then x = [ T ji ( a ) , Z ji ( b, r )] = T ji ( a ) · Z ji ( b,r ) T ji ( − a ) = T ji ( a ) · Z ji ( b,r ) [ T jh (1) , T hi ( − a )] . Thus, x = T ji ( a )[ Z ji ( b,r ) T jh (1) , Z ji ( b,r ) T hi ( − a )] = T ji ( a )[ T jh (1 − br ) T ih ( − rbr ) , T hj ( − arbr ) T hi ( − a (1 − rb ))] = T ji ( a )[ T jh (1) y, T hi ( − a ) z ] , where y = T jh ( − br ) T ih ( − rbr ) ∈ FU(2 n, J, ∆) ,z = T hj ( − arbr ) T hi ( arb ) ∈ FU(2 n, ( I, Γ) ◦ ( J, ∆)) . Since T hi ( − a ) ∈ FU(2 n, I,
Γ), the second factor of the above commutator belongs toFU(2 n, I,
Γ). Thus,[ T jh (1) y, T hi ( − a ) z ] = T jh (1) [ y, T hi ( − a ) z ] · [ T jh (1) , T hi ( − a ) z ] . (1)Now, the first commutator on the right hand side T jh (1) [ y, T hi ( − a ) z ] = T jh (1) [ T jh ( − br ) T ih ( − rbr ) , T hi ( − a ) T hj ( − arbr ) T hi ( arb )] . Expanding the commutator above by its second argument, we obtain T jh (1) [ T jh ( − br ) T ih ( − rbr ) , T hi ( − a ) T hj ( − arbr ) T hi ( arb )]= T jh (1) [ T jh ( − br ) T ih ( − rbr ) , T hi ( − a )] T jh (1) T hi ( − a ) [ T jh ( − br ) T ih ( − rbr ) , T hj ( − arbr ) T hi ( arb )] . The second factor above belongs to EU(2 n, ( I, Γ) ◦ ( J, ∆)). And the first factor aboveequals T jh (1) T jh ( − br ) [ T ih ( − rbr ) , T hi ( − a )] · [ T jh ( − br ) , T hi ( − a )]= T jh (1) T jh ( − br ) [ T ih ( − rbr ) , T hi ( − a )] · T ji ( bra ) ∈ T jh (1) T jh ( − br ) [ T ih ( − rbr ) , T hi ( − a )] · EU(2 n, ( I, Γ) ◦ ( J, ∆)) . On the other hand, the second commutator of (1) equals[ T jh (1) , T hi ( − a )] · T hi ( − a ) [ T jh (1) , z ] . The second commutator in the last expression belongs to EU(2 n, ( I, Γ) ◦ ( J, ∆)),and remains there after elementary conjugations, while the first commutator equals T ij ( − a ). Summarising the above, we see that x ∈ T ji ( a ) T jh (1) T jh ( − br ) [ T ih ( − rbr ) , T hi ( − a )] · EU(2 n, ( I, Γ) ◦ ( J, ∆))which belongs to [FU(2 n, I, Γ) , FU(2 n, J, ∆)] by Lemma 6. (cid:3)
Lemma 9.
Let ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) , ( J, ∆) be form ideals of ( A, Λ) . Suppose that a ∈ Γ , b ∈ ∆ and r ∈ Λ . Then [ T − i,i ( a ) , Z − i,i ( b, r )] ∈ [FU(2 n, I, Γ) , FU(2 n, J, ∆)] . Proof.
Without loss of generality, we may assume that i >
0. Pick an h > h = i . Then x = [ T − i,i ( a ) , Z − i,i ( b, r )] = T − i,i ( a ) · Z − i,i ( b,r ) T − i,i ( − a ) = T − i,i ( a ) · Z − i,i ( b,r ) (cid:0) T hi ( − a ) · [ T h, − h ( a ) , T − h,i (1)] (cid:1) . Thus, x = T − i,i ( a ) · (cid:0) Z − i,i ( b,r ) T hi ( − a ) · [ T h, − h ( a ) , Z − i,i ( b,r ) T − h,i (1)] (cid:1) = T − i,i ( a ) · T h,i ( − a (1 − br )) · T i, − h ( λrbra ) · (cid:2) T h, − h ( a ) , T − h,i (1 − rb ) · T i,h ( λrbr ) (cid:3) Using additivity of root unipotents, we can rewrite this as x = T − i,i ( a ) T h,i ( − a ) · T h,i ( − abr ) T i, − h ( λrbra ) · (cid:2) T h, − h ( a ) , T − h,i (1) T − h,i ( − rb ) · T i,h ( λrbr ) (cid:3) . Clearly, T h,i ( − abr ) T i, − h ( λrbra ) ∈ EU(2 n, ( I, Γ) ◦ ( J, ∆)) . On the other hand, the commutator in the last expression equals (cid:2) T h, − h ( a ) , T − h,i (1) T − h,i ( − rb ) · T i,h ( λrbr ) (cid:3) = (cid:2) T h, − h ( a ) , T − h,i (1) (cid:3) · T − h,i (1) (cid:2) T h, − h ( a ) , T − h,i ( − rb ) · T i,h ( λrbr ) (cid:3) = T h,i ( a ) T − i,i ( − a ) · T − h,i (1) (cid:2) T h, − h ( a ) , T − h,i ( − rb ) · T i,h ( λrbr ) (cid:3) . Again, clearly (cid:2) T h, − h ( a ) , T − h,i ( − rb ) · T i,h ( λrbr ) (cid:3) ∈ [FU(2 n, I, Γ) , FU(2 n, J, ∆)] . On the other hand, the previous factors assemble to a left T − i,i ( a ) T h,i ( − a ) conjugate ofan element of EU(2 n, ( I, Γ) ◦ ( J, ∆)) , which is contained in [FU(2 n, I, Γ) , FU(2 n, J, ∆)].This proves Lemma 9. (cid:3)
Combined, these results imply the first claim of Theorem 1.
OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 19 Rolling over elementary commutators
Now we pass to the final, and most difficult part of the proof of Theorem 1, rollingan elementary commutator over to a different position. Since we assume n ≥
3, thecase of short root type elementary commutators is easy. It is settled by essentiallythe same calculation as for the general linear group GL( n, R ), n ≥
3, see [76, 78]. Butfor the case of long root type elementary commutators we have to imitate the proofof [80], Theorems 4 and 5, for Sp(4 , R ). In the presence of non-trivial involution,non-commutativity and non-trivial form parameters this is quite a challenge. In § Lemma 10.
Let ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) , ( J, ∆) be form ideals of ( A, Λ) . Then for any i = ± j , any h = ± l , and any a ∈ I , b ∈ J , c , c ∈ A , one has Y ij ( c ac , b ) ≡ Y hl ( a, c bc ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Proof.
Take any h = ± i, ± j , and rewrite the elementary commutator z = Y ij ( c ac , b )on the left hand side of the above congruence as follows z = [ T ij ( c ac ) , T ji ( b )] = T ij ( c ac ) · T ji ( b ) T ij ( − c ac ) = T ij ( c ac ) · T ji ( b ) [ T hj ( ac ) , T ih ( c )] . Expanding the conjugation by T ji ( b ), we see that z = T ij ( c ac ) · [ T ji ( b ) T hj ( ac ) , T ji ( b ) T ih ( c )] = T ij ( c ac ) · (cid:2) [ T ji ( b ) , T hj ( ac )] T hj ( ac ) , T ih ( c )[ T ih ( − c ) , T ji ( b )] (cid:3) = T ij ( c ac ) · (cid:2) T hi ( − ac b ) T hj ( ac ) , T ih ( c ) T jh ( bc ) (cid:3) . Now, the first factor T hi ( − ac b ) of the first argument in this last commutator alreadybelongs to the group FU(2 n, ( I, Γ) ◦ ( J, ∆)). Thus, as above, z ≡ T ij ( c ac ) · (cid:2) T hj ( ac ) , T ih ( c ) T jh ( bc ) (cid:3) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Using multiplicativity of the commutator w.r.t. the second argument, cancelling thefirst two factors of the resulting expression, and then applying Lemma 6 we see that z ≡ T ih ( c ) [ T hj ( ac ) , T jh ( bc )] ≡ [ T hj ( ac ) , T jh ( bc )] (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . On the other hand, choosing another index l = ± j, ± h and rewriting the commu-tator [ T hj ( ac ) , T jh ( bc )] on the right hand side of the last congruence as[ T hj ( ac ) , T jh ( bc )] = [[ T hl ( a ) , T lj ( c )] , T jh ( bc )] , by the same argument we get the congruence z ≡ [ T hj ( ac ) , T jh ( bc )] ≡ [ T hl ( a ) , T lh ( c bc )] (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Obviously, for n ≥ i, j ), i = j , to any other suchposition ( k, m ), k = ± m , by a sequence of at most three such elementary moves. (cid:3) Lemma 11.
Let ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) , ( J, ∆) be form ideals of ( A, Λ) . Then for any − n ≤ i ≤ n , any − n ≤ k ≤ n , and any a ∈ λ − ( ε ( i )+1) / Γ , b ∈ λ ( ε ( i ) − / ∆ , c ∈ A , one has Y i, − i ( cac, b ) ≡ Y k, − k ( λ ( ε ( i ) − ε ( k )) / a, − λ ( ε ( k ) − ε ( i )) / cbc ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Proof.
Rewrite the elementary commutator z = Y i, − i ( cac, b ) on the left hand side ofthe above congruence as follows z = T i, − i ( cac ) · T − i,i ( b ) T i, − i ( − cac ) = T i, − i ( cac ) · T − i,i ( b ) (cid:0) T i, − k ( λ ( ε ( i ) − ε ( k )) / ca )[ T i,k ( c ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a )] (cid:1) . Expanding the conjugation by T − i,i ( b ), we see that z = T i, − i ( cac ) · T − i,i ( b ) T i, − k ( λ ( ε ( i ) − ε ( k )) / ca ) · (cid:2) T − i,i ( b ) T i,k ( c ) , T − i,i ( b ) T k, − k ( − λ ( ε ( i ) − ε ( k )) / a ) (cid:3) . Clearly, the first two factors y = T i, − i ( cac ) · T − i,i ( b ) T i, − k ( λ ( ε ( i ) − ε ( k )) / ca )can be rewritten as y = T i, − i ( cac ) · [ T − i,i ( b ) , T i, − k ( λ ( ε ( i ) − ε ( k )) / ca )] · T i, − k ( λ ( ε ( i ) − ε ( k )) / ca )which gives us the following congruence y ≡ T i, − i ( cac ) T i, − k ( λ ( ε ( i ) − ε ( k )) / ca ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . On the other hand, the commutator u = (cid:2) T − i,i ( b ) T i,k ( c ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a ) (cid:3) in the expression of z equals u = (cid:2) T − i,k ( bc ) T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc ) T i,k ( c ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a ) (cid:3) . Expanding this last expression, we get u = x [ T i,k ( c ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a )] · y [ T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a )] · [ T − i,k ( bc ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a )] , where x = T − i,k ( bc ) T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc ) , y = T − i,k ( bc ) . It is easy to see that[ T − i,k ( bc ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a )] ∈ EU(2 n, ( I, Γ) ◦ ( J, ∆)) , so we can drop it. Further, by Lemma 7, modulo EU(2 n, ( I, Γ) ◦ ( J, ∆)) the secondfactor can be simplified as follows y [ T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a )] ≡ [ T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a )] (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 21
But by Theorem 10 one has[ T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc ) , T k, − k ( − λ ( ε ( i ) − ε ( k )) / a )] ≡ [ T k, − k ( λ ( ε ( i ) − ε ( k )) / a ) , T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc )] (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Summarising the above, we get z ≡ T i, − i ( a ) T i, − k ( λ ( ε ( k ) − ε ( i )) / ca ) · x [ T i,k ( c ) , T k, − k ( − λ ( ε ( k ) − ε ( i )) / a )] · [ T k, − k ( λ ( ε ( i ) − ε ( k )) / a ) , T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc )] (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Thus, to finish the proof it suffices to show that v = T i, − i ( a ) T i, − k ( λ ( ε ( k ) − ε ( i )) / ca ) · x [ T i,k ( c ) , T k, − k ( − λ ( ε ( k ) − ε ( i )) / a )]belongs to EU(2 n, ( I, Γ) ◦ ( J, ∆)). Clearly, v = T i, − i ( cac ) T i, − k ( λ ( ε ( k ) − ε ( i )) / ca ) · x T i, − k ( − λ ( ε ( k ) − ε ( i )) / ca ) T i, − i ( − cac ) , can be rewritten as v = [ T i, − i ( cac ) T i, − k ( λ ( ε ( k ) − ε ( i )) / ca ) , x ] =[ T i, − i ( cac ) T i, − k ( λ ( ε ( k ) − ε ( i )) / ca ) , T − i,k ( bc ) T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc )] . Expanding this last commutator w.r.t. its first and second arguments, we express itas the product of elementary conjugates of the four following commutators • [ T i, − i ( cac ) , T − i,k ( bc )], • [ T i, − i ( cac ) , T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc )], • [ T i, − k ( λ ( ε ( k ) − ε ( i )) / ca ) , T − i,k ( bc )], • [ T i, − k ( λ ( ε ( k ) − ε ( i )) / ca ) , T − k,k ( − λ ( ε ( k ) − ε ( i )) / cbc )].A direct computation convinces us that each of these commutators belongs to theelementary subgroup EU(2 n, ( I, Γ) ◦ ( J, ∆)). This finishes the proof of lemma, andthus also of Theorem 1. (cid:3) Mat[ch]ing elementary commutators of different root lengths
In this section we prove a congruence connecting elementary commutators of longroot type with those of short root type. In the case, where one of the relativeform parameters is as small as possible (=minimal), this congruence can be used toeliminate long root type elementary commutators. On the other hand when one ofthe relative form parameters is as large as possible (=equals the corresponding ideal),one can abandon short root type elementary commutators.
Lemma 12.
Let ( A, Λ) be an associative form ring with , n ≥ , and let ( I, Γ) , ( J, ∆) be form ideals of ( A, Λ) . Then for any − n ≤ i ≤ n , any − n ≤ k ≤ n , and a ∈ I , b ∈ λ ( ε ( i ) − / ∆ , one has (cid:2) T i, − i ( a − λ ε ( − i ) a ) , T − i,i ( b ) (cid:3) ≡ [ T i,k ( a ) , T k,i ( b )] (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Proof.
Pick an index k = ± i , and rewrite the elementary commutator z = (cid:2) T i, − i ( a − λ ε ( − i ) a ) , T − i,i ( b ) (cid:3) on the left hand side as z = (cid:2) [ T k, − i ( − , T i,k ( a )] , T − i,i ( b ) (cid:3) = (cid:2) T k, − i ( − T i,k ( a ) · T i,k ( − a ) , T − i,i ( b ) (cid:3) . Using multiplicativity of the commutator w.r.t the first argument, we see z = T k, − i ( − T i,k ( a ) T k, − i (1) [ T i,k ( − a ) , T − i,i ( b )] · (cid:2) T k, − i ( − T i,k ( a ) , T − i,i ( b ) (cid:3) . The first factor belongs to EU(2 n, ( I, Γ) ◦ ( J, ∆)), so we leave it out. Thus, z iscongruent modulo this subgroup to (cid:2) T k, − i ( − T i,k ( a ) , T − i,i ( b ) (cid:3) = T k, − i ( − (cid:2) T i,k ( a ) , T k, − i (1) T − i,i ( b ) (cid:3) == T k, − i ( − (cid:2) T i,k ( a ) , [ T k, − i (1) , T − i,i ( b )] T − i,i ( b ) (cid:3) = T k, − i ( − (cid:2) T i,k ( a ) , T k,i ( b ) T k, − k ( λ ( ε ( − i ) − ε ( k )) / ( b )) T − i,i ( b ) (cid:3) . Expanding this last commutator w.r.t the second argument, we see that the secondand the third factors belong to EU(2 n, ( I, Γ) ◦ ( J, ∆)), so that we can leave them out.Now we have z ≡ T k, − i ( − (cid:2) T i,k ( a ) , T k,i ( b ) (cid:3) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) , as claimed. (cid:3) Corollary 1.
In conditions of Lemma further assume that b = b ′ − λ ε ( i ) b ′ for some b ′ ∈ J , then (cid:2) T i, − i ( a − λ ε ( − i ) a ) , T − i,i ( b − λ ε ( i ) b ) (cid:3) ≡ [ T i,k ( a ) , T k,i ( b ′ )] · [ T i,k ( a ) , T k,i ( − λ ε ( i ) b ′ )] modulo EU(2 n, ( I, Γ) ◦ ( J, ∆)) .Proof. Keep the notation from the proof of Lemma 12. Under this additional as-sumption one has z ≡ T k, − i ( − (cid:2) T i,k ( a ) , T k,i ( b ′ ) T k,i ( − λ ε ( i ) b ′ ) (cid:3) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . Expanding the commutator w.r.t the second argument again, we see that T k, − i ( − (cid:2) T i,k ( a ) , T k,i ( b ′ ) T k,i ( − λ ε ( i ) b ′ ) (cid:3) = T k, − i ( − (cid:0) [ T i,k ( a ) , T k,i ( b ′ )] · T k,i ( b ′ ) [ T i,k ( a ) , T k,i ( − λ ε ( i ) b ′ )] (cid:1) . Applying Lemma 6, we get z ≡ [ T i,k ( a ) , T k,i ( b ′ )] · [ T i,k ( a ) , T k,i ( − λ ε ( i ) b ′ )] (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) , as claimed. (cid:3) Corollary 2. If I = Γ or J = ∆ then for the second type of generators in Theorem it suffices to take one pair ( h, − h ) . Corollary 3. If Γ = I ∩ Λ min or ∆ = J ∩ Λ min then for the second type of generatorsin Theorem it suffices to take one pair ( h, k ) , h = ± k . OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 23 Triple and quadruple commutators
Actually Theorem 7 easily follows by induction on m from the following two specialcases, triple commutators, and quadruple commutators. Lemma 13.
Let ( A, Λ ) be any associative form ring with , let n ≥ , and let ( I, Γ) , ( J, ∆) , ( K, Ω) , be form ideals of ( A, Λ) . Then [[ EU(2 n, I, Γ) , EU(2 n, J, ∆)] , EU(2 n, K,
Ω)] =[ EU(2 n, ( I, Γ) ◦ ( J, ∆)) , EU(2 n, K,
Ω)] . Proof.
First of all, observe that the generators of the first type in Theorem 1 belong toEU(2 n, ( I, Γ) ◦ ( J, ∆)). Thus, forming their commutators with T h,k ( c ) ∈ EU(2 n, K,
Ω)will bring us inside [EU(2 n, ( I, Γ) ◦ ( J, ∆)) , EU(2 n, K,
Ω)].Next, let Y i,j ( a, b ) = [ T i,j ( a ) , T j,i ( b )] a typical generator of the second type of thecommutator subgroup [ EU(2 n, I, Γ) , EU(2 n, J, ∆)] with T i,j ( a ) ∈ EU(2 n, I,
Γ) and T j,i ( b ) ∈ EU(2 n, J, ∆).From Lemma 6 and Lemma 7 we know that x Y i,j ( a, b ) = Y i,j ( a, b ) z , for some z ∈ EU(2 n, ( I, Γ) ◦ ( J, ∆)), and thus for any T h,k ( c ) ∈ EU(2 n, K,
Ω),[ x Y i,j ( a, b ) , T k,l ( c )] = [ Y i,j ( a, b ) z, T k,l ( c )] = Y ij ( a,b ) [ z, T k,l ( c )] · [ Y i,j ( a, b ) , T k,l ( c )] . The first of these commutators also belongs to[ EU(2 n, ( I, Γ) ◦ ( J, ∆)) , EU(2 n, K,
Ω)] , and stays there after elementary conjugations. Let us concentrate at the second one. Case 1.
When i = ± j the same analysis as in the proof of Lemma 6, shows that: • If k = − l and k, l = ± i, ± j , then T k,l ( c ) commutes with Y i,j ( a, b ). • For any h = ± i, ± j the formulas for Y ij ( a, b ) and Y ij ( a, b ) − given in the proof ofLemma 6 immediately imply that[ z, T ih ( c )] = T jh ( babc ) T ih ( abc + ababc ) , [ z, T jh ( c )] = T jh ( − bac ) T ih ( − abac ) , [ z, T hi ( c )] = T jh ( caba ) T ih ( − cab ) , [ z, T hj ( c )] = T jh ( cba + cbaba ) T ih ( − cbab ) , and similarly[ z, T − i,h ( c )] = [ z, T − h,i ( − λ (( ε ( h )+ ε ( i )) / c )] = T j, − h ( λ (( ε ( h )+ ε ( i )) / caba ) T i, − h ( λ (( ε ( h )+ ε ( i )) / cab ) , [ z, T − j,h ( c )] = [ z, T − h,j ( − λ (( ε ( h )+ ε ( j )) / c )] = T j, − h ( λ (( ε ( h )+ ε ( j )) / cba + λ (( ε ( h )+ ε ( j )) / cbaba ) T i, − h ( λ (( ε ( h )+ ε ( j )) / cbab ) , [ z, T h, − i ( c )] = [ z, T i, − h ( − λ ( − ( ε ( i ) − ε ( h )) / c )] == T j, − h ( λ ( − ( ε ( i ) − ε ( h )) / bac ) T i, − h ( λ ( − ( ε ( i ) − ε ( h )) / abac ) , [ z, T h, − j ( c )] = [ z, T j, − h ( − λ ( − ( ε ( j ) − ε ( h )) / c )] T j, − h ( − λ (( ε ( j ) − ε ( h )) / bac ) T i, − h ( λ ( − ( ε ( j ) − ε ( h )) / abac )All factors on the right hand side belong already to EU (2 n, (( I, Γ) ◦ ( J, ∆)) ◦ ( K, Ω)).If ( k, l ) = ( ± i, ± j ) or ( ± j, ± i ), then we take an index h = ± i, ± j and rewrite T kl ( c )as [ T k,h ( c ) , T h,l (1)] and apply the previous items to get it belongs to [ EU(2 n, ( I, Γ) ◦ ( J, ∆)) , EU(2 n, K,
Ω)].On the other hand, for k = − l we have: • If k = ± i, ± j , then T k, − k ( c ) commutes with z and can be discarded. • Otherwise, we have[ z, T i, − i ( c )] = T i, − j ( − λ (( ε ( i ) − ε ( j )) / (1 + ab + abab ) c ( bab )) T j, − j ( λ (( ε ( i ) − ε ( j )) / babcbab ) T i, − i ( − c + (1 + ab + abab ) c (1 + ab + abab )) , [ z, T j, − j ( c )] = T i, − j ( abac (1 − ba )) T i, − i ( − λ (( ε ( j ) − ε ( i )) / abacaba ) · T j, − j ( − c + (1 − ba ) c (1 − ba )) , [ z, T − i,i ( c )] = [[ T ij ( a ) , T ji ( b )] , T − i,i ( c )] =[[ T − j, − i ( − λ (( ε ( j ) − ε ( i )) / a ) , T − i, − j ( λ (( ε ( i ) − ε ( j )) / b )] , T − i,i ( c )] , [ z, T − j,j ( c )] = [[ T ij ( a ) , T ji ( b )] , T − j,j ( c )] =[[ T − j, − i ( − λ (( ε ( j ) − ε ( i )) / a ) , T − i, − j ( λ (( ε ( i ) − ε ( j )) / b )] , T − j, − j ( c )] . The two last cases reduce to the first two. In each case the resulting expressionsbelong to EU (2 n, (( I, Γ) ◦ ( J, ∆)) ◦ ( K, Ω)).
Case 2.
When i = − j the same analysis as in the proof of Lemma 7, shows that: • If ( k, l ) = ( − i, i ), then[ z, T − i,i ( c )] = [[ T i, − i ( a ) , T − i,i ( b )] , T − i,i ( c )] = [ Z − i,i ( b, a ) , T − i,i ( c )] . Now, the same computation as in Lemma 9 shows that[ z, T − i,i ( c )] ∈ EU(2 n, ( I, Γ) ◦ ( J, ∆)) . OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 25 • If ( k, l ) = ( i, − i ), then[ z, T i, − i ( c )] = [[ T i, − i ( a ) , T − i,i ( b )] , T i, − i ( c )] = [[ T − i,i ( b ) , T i, − i ( a )] − , T i, − i ( c )]= [ T − i,i ( b ) , T i, − i ( a )] · [ T i, − i ( c ) , [ T − i,i ( b ) , T i, − i ( a )]] · [ T − i,i ( b ) , T i, − i ( a )] − . By the previous subcase,[ T i, − i ( c ) , [ T − i,i ( b ) , T i, − i ( a )]] ∈ EU(2 n, ( I, Γ) ◦ ( J, ∆)) . But then its conjugates also stay therein. • If k = i and j = ± k , then[ z, T i,j ( c )] = [[ T i, − i ( a ) , T − i,i ( b )] , T i,j ( c )] = T − j,j ( λ (( ε ( j ) − ε ( i )) / cbabcλ ε ( j ) ( cbababc + cbabababc )) · T − i,j ( babc ) T i,j (( ab + abab ) c )Since a ∈ λ − ( ε ( i )+1) / Γ and b ∈ λ ( ε ( i ) − / ∆, it follows that the right hand side belongsto EU(2 n, ( I, Γ) ◦ ( J, ∆)). • If k = − i and j = ± k , then[ z, T − i,j ( c )] = [[ T i, − i ( a ) , T − i,i ( b )] , T − i,j ( c )] =[ T − i,i ( b ) , T i, − i ( a )] − · [ T − i,j ( c ) , [ T − i,i ( b ) , T i, − i ( a )]] · [ T − i,i ( b ) , T i, − i ( a )] . By the previous subcase,[ T − i,j ( c ) , [ T − i,i ( b ) , T i, − i ( a )]] ∈ EU(2 n, ( I, Γ) ◦ ( J, ∆)) . But then its conjugates also stay therein. • Finally, using relation ( R
1) the subcase l = ± i and k = ± i is readily reduced tothe subcases, where k = ± i . (cid:3) Now, for n ≥ n = 3 it requires a separate proof. All our assaultson this remaining case were crippled by forbidding calculations. Lemma 14.
Let ( A, Λ ) be any associative form ring with and let ( I, Γ) , ( J, ∆) , ( K, Ω) , ( L, Θ) be form ideals of ( A, Λ) . If either n ≥ or there exists an ideal equalsits corresponding relative form parameter and n ≥ , then (cid:2) [ EU(2 n, I, Γ) , EU(2 n, J, ∆)] , [ EU(2 n, K, Ω) , EU(2 n, L,
Θ)] (cid:3) =[ EU(2 n, ( I, Γ) ◦ ( J, ∆)) , EU(2 n, ( K, Ω) ◦ ( L, Θ))] . Proof.
From the previous lemma we already know that (cid:2)
EU(2 n, ( I, Γ) ◦ ( J, ∆)) , [ EU(2 n, K, Ω) , EU(2 n, L,
Θ)] (cid:3) = (cid:2) EU(2 n, ( I, Γ) ◦ ( J, ∆)) , EU(2 n, ( K, Ω) ◦ ( L, Θ)) (cid:3) and that (cid:2) [ EU(2 n, I, Γ) , EU(2 n, J, ∆)] , EU(2 n, ( K, Ω) ◦ ( L, Θ)) (cid:3) = (cid:2) EU(2 n, ( I, Γ) ◦ ( J, ∆)) , EU(2 n, ( K, Ω) ◦ ( L, Θ)) (cid:3) . Thus, it only remains to prove that[ Y ij ( a, b ) , Y hk ( c, d )] ∈ (cid:2) EU(2 n, ( I, Γ) ◦ ( J, ∆)) , EU(2 n, ( K, Ω) ◦ ( L, Θ)) (cid:3) , where a ∈ ( I, Γ), b ∈ ( J, ∆), c ∈ ( K, Ω) and d ∈ ( L, Θ). Conjugations by elements x ∈ EU(2 n, A,
Λ) do not matter, since they amount to extra factors from the abovetriple commutators, which are already accounted for.Now, for n ≥ Y hk ( c, d ) modulo EU(2 n, ( K, Ω) ◦ ( L, Θ)) to a position, where it commutes with Y ij ( a, b )], either by Lemma 10 when i = ± j and h = ± k or by Lemma 11 when i = − j or h = − k .Suppose that there exists an ideal equals its corresponding relative form paramerter,say I = Γ. If i = ± j then by Lemma 12, we have Y i,j ( a, b ) ≡ Y i, − i ( a, b − λ ε ( i ) b ) . For n ≥
3, we can move Y i, − i ( a, b − λ ε ( i ) b ) module EU(2 n, ( K, Ω) ◦ ( L, Θ)) to a position,where it commutes with Y hk ( c, d ) by Lemma 10. Otherwise, if i = − j then can alsomove Y i, − i ( a, b ) to a position, where it commutes with Y hk ( c, d ) by Lemma 11. Thisfinishes the whole proof. (cid:3) Elementary multiple commutator formulas
In the current section, we show that multiple commutators of elementary subgroupscan be reduced to double such commutators.To state our main results, we have to recall some further pieces of notation from[22, 33, 23, 31, 27, 64]. Namely, let H , . . . , H m ≤ G be subgroups of G . There aremany ways to form a higher commutator of these groups, depending on where weput the brackets. Thus, for three subgroups F, H, K ≤ G one can form two triplecommutators [[ F, H ] , K ] and [ F, [ H, K ]]. Usually, we write [ H , H , . . . , H m ] for the left-normed commutator, defined inductively by[ H , . . . , H m − , H m ] = [[ H , . . . , H m − ] , H m ] . To stress that here we consider any commutator of these subgroups, with an arbitraryplacement of brackets, we write J H , H , . . . , H m K . Thus, for instance, J F, H, K K refersto any of the two arrangements above.Actually, a specific arrangement of brackets usually does not play major role in ourresults – apart from one important attribute . Namely, what will matter a lot is the Actually, for non-commutative rings symmetric product of ideals is not associative, so thatthe initial bracketing of higher commutators will be reflected also in the bracketing of such highersymmetric products.
OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 27 position of the outermost pairs of inner brackets. Namely, every higher commutatorsubgroup J H , H , . . . , H m K can be uniquely written as J H , H , . . . , H m K = [ J H , . . . , H s K , J H s +1 , . . . , H m K ] , for some s = 1 , . . . , m −
1. This s will be called the cut point of our multiplecommutator. Now we are all set to finish the proof of Theorem 7. The proof is aneasy adaptation of the proof of [78], Theorem 1, but we reproduce it here for the sakeof completeness. Proof.
Denote the commutator on the left-hand side by H , H = J EU(2 n, I , Γ ) , EU(2 n, I , Γ ) , . . . , EU(2 n, I m , Γ m ) K . We argue by induction in m , with the cases m ≤ m = 2 there is nothing to prove, case m = 3 is accounted for by Lemma 13, andcase m = 4 — by Lemma 13, if the cut point s = 2, and by Lemma 14 when s = 2.Now, let m ≥ . . . ]] with the cut point s and let J EU(2 n, I , Γ ) , EU(2 n, I , Γ ) , . . . , EU(2 n, I s , Γ s ) K , J EU(2 n, I s +1 , Γ s +1 ) , EU(2 n, I s +2 , Γ s +2 ) , . . . , EU(2 n, I m , Γ m ) K , be the partial commutators, the first one containing the factors afore the cut point,and the second one containing those after the cut point. • When the cut point occurs at s = 1 or at s = m −
1, one of these commutatorsis a single elementary subgroup EU(2 n, I ) in the first case or EU(2 n, I m − ) in thesecond one. Then we can apply the induction hypothesis to another factor. For s = 1,denote by t = 2 , . . . , m − H = h EU(2 n, I , Γ ) , q EU(2 n, I , Γ ) , EU(2 n, I , Γ ) , . . . , EU(2 n, I m , Γ m ) y i = h EU(2 n, I , Γ ) , (cid:2) EU(2 n, ( I , Γ ) ◦ . . . ◦ ( I t , Γ t )) , EU(2 n, ( I t +1 , Γ t +1 ) ◦ . . . ◦ ( I m , Γ m )) (cid:3)i , and we are done by Lemma 13. Similarly, for s = m − r = 1 , . . . , m − H = h q EU(2 n, I , Γ ) , EU(2 n, I , Γ ) , . . . , EU(2 n, I m − , Γ m − ) y , EU(2 n, I m , Γ m ) i = h(cid:2) EU(2 n, ( I , Γ ) ◦ . . . ◦ ( I r , Γ r )) , EU(2 n, ( I r +1 , Γ r +1 ) ◦ . . . ◦ ( I m − , Γ m − )) (cid:3) , EU(2 n, I m , Γ m ) i , and we are again done by Lemma 13. • Otherwise, when s = 1 , m −
1, we can apply the induction hypothesis to bothfactors. Let as above r = 1 , . . . , s − t = s + 1 , . . . , m − H = h q EU(2 n, I ) , EU(2 n, I ) , . . . , EU(2 n, I s ) y , q EU(2 n, I s +1 ) , EU(2 n, I s +2 ) , . . . , EU(2 n, I m ) y i to conclude that H = h(cid:2) EU(2 n, I ◦ . . . ◦ I r ) , EU(2 n, I r +1 ◦ . . . ◦ I s ) (cid:3) , (cid:2) EU(2 n, I s +1 ◦ . . . ◦ I t ) , EU(2 n, I t +1 ◦ . . . ◦ I m ) (cid:3)i , and we are again done, this time by Lemma 14. (cid:3) Further applications
Now, we are in a position to finish the proof of Theorem 8.
Proof.
Since ( I, Γ) and ( J, ∆) are comaximal, there exist a ′ ∈ I and b ′ ∈ J such that a ′ + b ′ = 1 ∈ R . But then by Lemmas 10 and 12, for i = ± j one has Y ij ( a, b ) = Y ij ( a ( a ′ + b ′ ) , b ) ≡ Y ij ( aa ′ , b ) · Y ij ( ab ′ , b ) ≡ e modulo EU(2 n, ( I, Γ) ◦ ( J, ∆)).For i = − j , one has Y i, − i ( a, b ) = Y i, − i (( a ′ + b ′ ) a ( a ′ + b ′ ) , b ) = Y i, − i ( a ′ aa ′ + b ′ aa ′ + a ′ ab ′ + b ′ ab ′ , b ) . Applying multiplicativity of commutators to the first argument of the above commu-tator and then Lemma 7, we deduce z ≡ Y i, − i ( a ′ aa ′ , b ) Y i, − i ( b ′ aa ′ , b ) Y i, − i ( a ′ ab ′ , b ) Y i, − i ( b ′ ab ′ , b ) (mod EU(2 n, ( I, Γ) ◦ ( J, ∆))) . By Theorem 10, each of above factors is trivial modulo EU(2 n, ( I, Γ) ◦ ( J, ∆)). Thisfinishes the proof. (cid:3) Let us state another amusing corollary of Theorem 10. For the form ideals them-selves, one has an obvious inclusion (cid:0) ( I, Γ) + ( J, ∆) (cid:1) ◦ (cid:0) ( I, Γ) ∩ ( J, ∆) (cid:1) = (cid:0) ( I + J ) ◦ ( I ∩ J ) , Γ min (( I + J ) ◦ ( I ∩ J )) + (Γ ∩ ∆) (Γ + ∆) + (Γ+∆) (Γ ∩ ∆) (cid:1) ≤ (cid:0) I ◦ J, Γ min ( I ◦ J ) + J Γ + I ∆ (cid:1) = ( I, Γ) ◦ ( J, ∆) . Only very rarely this inclusion is always an equality.
Theorem 11.
For any two form ideals ( I, Γ) and ( J, ∆) of ( A, Λ) , n ≥ , one has (cid:2) EU (2 n, ( I, Γ) + ( J, ∆)) , EU ( n, ( I, Γ) ∩ ( J, ∆)) (cid:3) ≤ [ EU(2 n, I, Γ) , EU(2 n, J, ∆)] . OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 29
Proof.
The observation immediately preceding the theorem shows that the level ofthe left hand side is contained in the level of the right hand side,EU (cid:0) n, R, (( I, Γ) + ( J, ∆)) ◦ (( I, Γ) ∩ ( J, ∆)) (cid:1) ≤ EU (2 n, R, ( I, Γ) ◦ ( J, ∆)) . Thus, it only remains to prove that the elementary commutators Y ij ( a + b, c ), with a ∈ ( I, Γ), b ∈ ( J, ∆), c ∈ ( I, Γ) ∩ ( J, ∆), in the left hand side belong to the righthand side.By Theorem 10, one has Y ij ( a + b, c ) ≡ Y ij ( a, c ) · Y ij ( b, c ) (mod EU (2 n, R, (( I, Γ) + ( J, ∆)) ◦ (( I, Γ) ∩ ( J, ∆)))) . Thus, this congruence holds also modulo the larger subgroup EU(2 n, R, ( I, Γ) ◦ ( J, ∆)).On the other hand, Theorem 6 implies that Y ij ( b, c ) ≡ Y ij ( c, − b ) (mod EU(2 n, R, ( I, Γ) ◦ ( J, ∆))) . Combining the above congruences, we see that Y ij ( a + b, c ) ≡ Y ij ( a, c ) · Y ij ( c, − b ) (mod EU(2 n, R, ( I, Γ) ◦ ( J, ∆))) , where both commutators in the right hand side belong to [EU(2 n, I, Γ) , EU(2 n, J, ∆)],which proves the desired inclusion. (cid:3)
Final remarks
Here we make some further observations concerning the context of this work andalso state some unsolved problems and reiterate some further problems from [27, 31],which are still pending.12.1.
How we got here.
The study of birelative standard commutator formulasgoes back to the foundational work by Hyman Bass [10]. As early successes oneshould also mention important contributions by Alec Mason and Wilson Stothers[42, 39, 40, 41] and by Hong You [84]. Our own research in this direction started in2008–2010 in the joint works with Alexei Stepanov and Roozbeh Hazrat [74, 32, 75]and was then continued in 2011–2017 in a series of our joint works based on relativeversions of localisation methods, in particular [33, 27, 28, 29, 30, 31]. Simultaneously,Stepanov developed his universal localisation and applied it to multiple commutatorformulas and commutator width, see [63, 64]. One can find systematic description ofthat stage of development in our surveys and conference papers [22, 24, 23, 30].The present work is a natural extension of our more recent papers [71, 76, 72, 77,78, 79, 80]. It owes its existence to the two following momentous observations wemade in October 2018, and in September 2019, respectively.In October 2018 the first author proved a special case of Theorems 2 and 3 for thegeneral linear group GL( n, R ), n ≥
3, over commutative rings, see [71]. The initial At least three our scheduled works of that period, which were essentially completed by 2016, viz.,the general multiple commutator formula for GL( n, R ), unitary commutator width, and analysis ofthe case GU(4 , R,
Λ), still remain unpublished. proof employed a version of decomposition of unipotents [65], that was already usedfor a similar purpose in his joint work with Alexei Stepanov [74]. The second authorthen immediately observed that Theorem 2 implies the first claim of Theorem 1 andthat it should be possible to proceed conversely, first establish a version of Theorem 1by elementary calculations, and then derive Theorems 2 and 3. This is exactly whatwas done for Chevalley groups in our paper [76], again over commutative rings.In July–September 2019 the first author was discussing bounded generation ofChevalley groups in the function case with Boris Kunyavsky and Eugene Plotkin.One of the tricks used in many published papers consisted in splitting an elementaryconjugate/elementary commutator and then reassembling it in a different position.We noticed that the same calculation of rolling elementary conjugates to a differentposition appeared over and over again in many different contexts: • Congruence subgroup problem.
In a preliminary mode it was already present inthe precursory article by Jens Mennicke [43] and then already in full-fledged form inthe epoch-making memoir by Hyman Bass, John Milnor, and Jean-Pierre Serre [12],behold the proof of Theorem 5.4. • Bounded generation . Post factum, we discerned the same calculation in theclassical papers by David Carter, Gordon Keller, and Oleg Tavgen [15, 68], but weonly became aware of that perusing a recent article by Bogdan Nica [44]. • In fact, Wilberd van der Kallen and Alexei Stepanov [34, 62, 63] use a verysimilar calculation to reduce the generating sets of relative elementary subgroups.Here we attached merely a handful of references. Retrospectively, we spotted thesame or very similar calculations in oodles of further papers, but apparently it washardly ever applied in the birelative context.At the end of September the first author used essentially the same calculation to prove that when R is commutative and n ≥ E ( n, A ) , E ( n, B )] is contained in another birelative groupEE( n, A, B ) = h t ij ( c ) , where c ∈ A, i < j, and c ∈ B, i > j i , see [72], Theorem 3. Within a few days of vehement correspondence we observedthat everything works over arbitrary associative rings and can be further enhancedto entail Theorems 1 and 5 for GL( n, R ). This is done in [76], and soon thereafter ina more mature form, implying also Theorems 6, 7 and 8, in [78].Morally, the present paper, and a parallel paper that addresses the case of Chevalleygroups [80], are direct offsprings of this development. However, technically these casesturned out to be way more demanding, and we had to spend quite some time to supplydetailed proofs of all auxiliary results. Simultaneously and independently exactly the same calculation was applied by Andrei Lavrenovand Sergei Sinchuk [38] at the level of K . OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 31
Degree improvements.
Of course, the first question that immediately occursis whether Theorem 7 holds also for n = 3. For quasi-finite rings this is indeed thecase [30], and we are pretty more inclined to believe in the positive answer. Problem 1.
Prove that Lemma and Theorem hold also for n = 3 . Getting a proof in the same style as that of Lemma 13 seems to be highly non-trivialfrom a technical viewpoint. However, the possibility to construct a counter-exampleappears even more remote.In the main body of the present paper we always assumed that n ≥
3. Obviously,due to the exceptional behavior of the orthogonal group SO(4 , A ), these results donot fully generalise to the case n = 2. It is natural to ask, whether results of thepresent paper hold also for the group GU(4 , A, Λ). However, this obviously fails ingeneral without some strong additional assumptions on the form ring and/or formideals.Still, we believe they do generalise, provided Λ A + A Λ = A , or the like. Knownresults clearly indicate both that this should be possible, and that the analysis ofthe case n = 2 will be considerably harder from a technical viewpoint, than that ofthe case n ≥ Problem 2.
Generalise results of the present paper to the group
GU(4 , A, Λ) , providedthat Λ A + A Λ = A , Γ J + J Γ = I , ∆ I + I ∆ = J , or the like. Actually, some 8 years ago we have obtained various headways towards the relativestandard commutator formula and all that for GU(4 , A,
Λ), but even these results areunpublished, due to their fiercely technical character.12.3.
Presentations and stability.
As a counterpart to Theorem 9 we can ask,whether the stability map for this quotient is also injective. A natural approach tothis would be to tackle the following much more ambitious project.
Problem 3.
Give a presentation of [ EU(2 n, I, Γ) , EU(2 n, J, ∆)] / EU(2 n, A, ( I, Γ) ◦ ( J, ∆)) by generators and relations. Does this presentation depend on n ≥ ? In Theorems 6 and 10 and Lemma 12 we have established some of the relationsamong the elementary commutators modulo EU(2 n, A, ( I, Γ) ◦ ( J, ∆)). However, easyarithmetic examples show this is not a defining set of relations, so that there mustbe some further relations. Compare [76, 78, 79] for discussion of the similar problemfor GL( n, A ). Compare the work by Bak and the first author [8], and references therein.
Higher relations.
In [79] we established some further congruences for theelementary commutators in GL( n, A ), n ≥
3, where A is an arbitrary associativering. The highlight of that paper is the following remarkable triple congruence, aversion of the Hall—Witt identity.Let I, J, K be two-sided ideals of R . Then for any three distinct indices i, j, h suchthat 1 ≤ i, j, h ≤ n , and all a ∈ I , b ∈ J , c ∈ K , one has y ij ( ab, c ) y jh ( ca, b ) y hi ( bc, a ) ≡ e (mod E ( n, R, IJ K + J KI + KIJ )) , see [79], Theorem 1. This identity has lots of applications, including many new inclu-sions among double and multiple mixed relative elementary commutator subgroups.Specifically, it allows to solve the analogue of Problem 3 for GL( n, A ) in the par-ticularly agreeable case of Dedekind rings. Thus, it would be most natural to seekout similar higher congruences in the unitary case as well. Problem 4.
Generalise the results of [79] to the unitary groups
GU(2 n, A, Λ) , n ≥ . One such congruence among short root type elementary commutators is immedi-ately clear. But the congruences involving long root type elementary commutatorswill be fancier and longer.12.5.
Other birelative groups.
Let us briefly discuss two further groups dependingon two form ideals of a form ring. First of all, it is the partially relativised groupFU(2 n, I, Γ) FU(2 n,J, ∆) . It seems that in view of the identityFU(2 n, I, Γ) FU(2 n,J, ∆) = [FU(2 n, I, Γ) , FU(2 n, J, ∆)] · FU(2 n, I, Γ) , our Theorem 1 readily implies the following generalisation of [9], Proposition 5.1,to FU(2 n, I, Γ) FU(2 n,J, ∆) . Namely, we assert that it is generated by the appropriateelementary conjugates. Problem 5.
Prove that the partially relativised groups
FU(2 n, I, Γ) FU(2 n,J, ∆) are gen-erated by T ji ( b ) T ij ( a ) , where a ∈ ( I, Γ) , b ∈ ( J, ∆) . Another birelative group EEU(2 n, ( I, Γ) , ( J, ∆)) is defined as followsEEU(2 n, ( I, Γ) , ( J, ∆)) = h T ij ( a ) , where c ∈ ( I, Γ) , i < j, and c ∈ ( J, ∆) , i > j i . The following problem proposes a unitary generalisation of [72], Theorem 3, wherea similar result was established for GL( n, A ). Problem 6.
Prove that [FU(2 n, I, Γ) , FU(2 n, J, ∆)] ≤ EEU(2 n, ( I, Γ) , ( J, ∆)) . General multiple commutator formula.
Let us now recall another majorunsolved problem as stated already in [27, 30] and [31], Problem 1. We proffer toprove general multiple commutator formula for unitary groups.
OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 33
Problem 7.
Let ( I i , Γ i ) , ≤ i ≤ m , be form ideals of the form ring ( A, Λ) such that A is module-finite over a commutative ring R that has finite BassSerre dimension δ ( R ) = d < ∞ . Prove that for any m ≥ d one has J GU(2 n, I , Γ ) , GU(2 n, I , Γ ) , . . . , GU(2 n, I m , Γ m ) K = J EU(2 n, I , Γ ) , EU(2 n, I , Γ ) , . . . , EU(2 n, I m , Γ m ) K . Observe that the arrangement of brackets in the above formula should be the sameon both sides as the mixed commutators are not associative. A similar problemfor algebraic groups over commutative rings, in particular for Chevalley groups, wassolved by Alexei Stepanov [64], by his remarkable universal localisation method.Recall that the proof of a similar result for GL( n, R ) over non-commutative rings isbased on the following result of Mason—Stothers [42], Theorem 3.6 and Corollary 3.9,see [30], Theorem 13, for an easy modern proof. Of course, that we can unrelativisethe right hand side was only established in [76], Theorem 2, so formally this theoremwas never stated in this form.
Theorem C.
Let A be a ring, I and J be two two-sided ideals of A . Assume that n ≥ sr( R ) , . Then [GL( n, A, I ) , GL( n, A, J )] = [ E ( n, I ) , E ( n, J )] . For unitary groups, even such basic facts at the stable level seem to be missing.
Problem 8.
Find appropriate stability conditions under which [GU(2 n, I, Γ) , GU(2 n, J, ∆)] = [FU(2 n, I, Γ) , FU(2 n, J, ∆)] . After that, the proof in our unpublished paper proceeds by induction on d , whichdepends on Baks results [3], precise form of injective stability for K , such as theBassVaserstein theorem, etc. It seems that to solve Problem 7 one has to rethink andexpand many aspects of structure theory of unitary groups, starting with stabilitytheorems for KU .The first complete generally accepted proof of injective stability for KU wasobtained (but not published!) by Maria Saliani [56], and first published by Max Knusin his book [35]. After that, generalisations and improvements were proposed byAnthony Bak, Guoping Tang, Victor Petrov, and Sergei Sinchuk [7, 5, 60], and thenvery recently by Weibo Yu, Rabeya Basu and Egor Voronetsky [87, 14, 82].Problem 7 is also intimately related to the nilpotent structure of KU . In theabsolute case the corresponding results for unitary groups were obtained by RoozbehHazrat in his Ph. D. Thesis [20, 21], and in the relative case in a joint paper by Bak, In late 1960-ies and mid 1970-ies Anthony Bak and Manfred Kolster obtained stability understronger assumptions, with very sketchy proofs. Leonid Vaserstein worked in smaller generalityas far as groups, and his proof of injective stability for unitary groups contained serious gaps andinaccuracies. In 1980 Mamed-Emin Oglu Namik Mustafa-Zadeh announced surjective stability forKU — and thus also injective stability for KU — in full generality. However, a complete proofwas never published, and the exposition in his 1983 Ph. D. Thesis is blurred by serious mistakes. Hazrat and the first author [4]. To fully cope with Problem 7, we need more powerfulresults on the superspecial unitary groups than what was established in [4]. Part ofwhat is demanded here was recently established by Weibo Yu, Guoping Tang andRabeya Basu [88, 13], but there is still a lot of work to be done.12.7.
Subnormal subgroups.
Initially, one of our main motivations to pursue thework on birelative commutator formulas were prospective applications to the studyof subnormal subgroups of GU(2 n, A,
Λ). As was observed by John Wilson [83],technically this amounts to description of subgroups of GU(2 n, A,
Λ), normalised bya relative elementary subgroup EU(2 n, J, ∆), for some form ideal ( J, ∆).A major early contribution is due to Gnter Habdank [17, 18], who additionallyassumed that the form ring was subject to some stability conditions. Definitiveresults for quasi-finite rings were then obtained by the second author and You Hong[90, 91, 92, 85]. However, we are convinced that the bounds in these papers can befurther improved and hope to return to the following problem with our new tools. Problem 9.
Obtain optimal bounds in the description of subgroups of
GU(2 n, A, Λ) ,normalised by the relative elementary subgroup EU(2 n, J, ∆) , for a form ideal ( J, ∆) E ( A, Λ) . Until recently, for the unitary groups the proofs of structure theorems were inbad shape even in the absolute case . However, now the situation has changed. In2013 Hong You and Xuemei Zhou [86] published a detailed proof for commutativeform rings. Finally, in 2014 Raimund Preusser in his Ph. D. Thesis [49] gave a firstcomplete localisation proof for quasi-finite form rings, which is published in [50].In 2017 Raimund Preusser [51, 52] has also finally succeeded in completing a globalproof as envisaged in [9]. These papers constitute a major breakthrough since, atleast for commutative rings, they give explicit polynomial expressions of non-trivialtransvections as products of elementary conjugates of a given matrix and its inverse.(See also [53, 55] for further results in this spirit for GL( n, A ) over various classes ofnon-commutative rings.) The first author has immediately recognised that the resultsby Preusser procure an effectivisation for the description of normal subgroups in muchthe same sense as the decomposition of unipotents [65], does for the normality of theelementary subgroup. This prompted him to call this method reverse decompositionof unipotents [70]. Moreover, he noticed that in the case of GL( n, A ) these results canbe generalised (with only marginally worse bounds) to the description of subgroupsnormalised by a relative elementary subgroups [73].We are confident that, combining the methods developed by Preusser in the abovepapers with our methods, we could easily improve bounds in all published results for As indicated in [26], the proof in the work by Leonid Vaserstein and Hong You [69] containeda major omission, and only established the weak structure theorem. The details of the purportedglobal proof by Bak and the first author, that was around since the early 1990-ies, and that washarbingered in [9], remained unpublished.
OMMUTATORS OF RELATIVE ELEMENTARY UNITARY GROUPS 35 unitary groups. Of course, to prove that the bounds thus obtained are themselvesthe best possible ones would be quite a challenge.12.8.
Commutator width.
Another related problem that initially motivated ourwork was the study of commutator width. Alexander Sivatsky and Alexei Stepanov[61] have discovered that over rings of finite Jacobson dimension j-dim( A ) = d < ∞ any commutator [ x, y ], where x ∈ GL( n, A ), y ∈ E ( n, A ), is a product of ≤ L elementary generators, where L = L ( n, d ) only depends on n and d . This result wasthen generalised to all Chevalley groups G (Φ , A ) by Stepanov and the first author[66], with the bound depending on the type Φ and on d .Ultimately, Stepanov discovered that for reductive groups similar results hold for arbitrary commutative rings and that the bound L therein depends on the type of thegroup alone and not on the ring A . Also, he discovered that similar results hold atthe relative and birelative level, with elementary conjugates and our generators (likethose in Theorem B) as the generating sets of [ E (Φ , A, I ) , E (Φ , A, J )], again withbounds that depend on the type alone, and not on A , I or J . See [24] for statementsand detailed discussion of these results.However, Bak’s unitary groups are not always algebraic and similar results oncommutator width are not yet published even in the absolute case and even overfinite-dimensional rings. Problem 10.
Let ( A, Λ) be a commutative form ring such that j-dim( A ) < ∞ . Provethat the length of commutators in [GU(Φ , A, I ) , E (Φ , A, J )] in terms of the generatorslisted in Theorem is bounded, and estimate this length. Alexei Stepanov maintained that the above length is bounded in the absolute case,without actually producing any specific bound. To obtain an exponential bounddepending on d by relative localisation methods [27, 31, 30] would be simply a matterof patience. Actually, this was essentially done by ourselves and Roozbeh Hazrat,but even in the absolute case all of this still remains unpublished.On the other hand, to achieve a uniform polynomial bound, similar to the oneestablished in [61] for GL( n, A ) but not depending on d , one would need to combinea full-scale generalisation of Stepanov’s universal localisation to unitary groups, withfull-scale unitary versions of decomposition of unipotents, including explicit poly-nomial formulae for the conjugates of root unipotents. This seems to be a ratherambitious project.12.9. Unitary Steinberg groups.
It is natural to ask to which extent our methodsand results carry over to the level of KU . Problem 11.
Prove analogues of the main results of the present paper for the unitarySteinberg groups
StU(2 n, A, Λ) . For the definition of unitary Steinberg groups see [2, 36] and references there (or[37] for odd unitary Steinberg groups). Here, we do not discuss subtleties related to the definition of relative unitary Steinberg groups, as also relation to excision inunitary algebraic K -theory, etc.12.10. Description of subgroups.
The methods of the present paper can haveapplications also in description of various classes of subgroups of unitary groups. Notin the position to discuss this at any depth here, we just cite the works by VictorPetrov, Alexander Shchegolev and Egor Voronetsky [46, 57, 58, 59, 81] where one canfind many further references. Observe that the result by Voronetsky [81] is especiallypowerful, since it simultaneously generalises also the description of EU-normalisedsubgroups (in the context of odd unitary groups!)12.11.
Odd unitary groups.
Finally, we are positive that all results of the presentpaper generalise also to odd unitary groups introduced by Victor Petrov [47, 48].
Problem 12.
Generalise the results of [27, 29, 30] and the present paper to oddunitary groups, under suitable isotropy assumptions.
Of course, this is not an individual clear-cut problem, but rather a huge researchproject. Clearly, in most cases the proofs in this setting will require much moreonerous calculations. Let us cite some important recent papers by Yu Weibo, TangGuoping, Li Yaya, Liu Hang, Anthony Bak, Raimund Preusser and Egor Voronetsky[88, 89, 6, 54, 81, 82] that address normal structure and stability for odd unitarygroups.12.12.
Acknowledgements.
We thank Anthony Bak, Roozbeh Hazrat and AlexeiStepanov for long-standing close cooperation on this type of problems over the lastdecades. The present paper gradualy evolved to the current shape between Decem-ber 2018 and March 2020. The first author thanks Boris Kunyavsky and EugenePlotkin, for ongoing discussion and comparison of the existing proofs of the congru-ence subgroup problem and bounded generation in terms of elementaries. The boutof these deliberations that has taken place on September 16, 2019, first in “BibliotekaCafe”, and then in “Manneken Pis” on Kazanskaya, was especially fateful for [72] andall subsequent development. We thank Pavel Gvozdevsky, Andrei Lavrenov, SergeiSinchuk and Anastasia Stavrova for their very pertinent questions and comments.We are extremely grateful also to Fan Huijun for his friendly support. In particular,he organised a visit of the first author to Peking University in December 2019, whichgave us an excellent opportunity to coordinate our vision.
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