aa r X i v : . [ m a t h . R A ] F e b TWO LIVES : COMPOSITIONS OF UNIMODULAR ROWS
VINEETH CHINTALA
Abstract.
The paper lays the foundation for the study of unimodularrows using Spin groups. We show that E n ( R )-orbits of unimodular rowsare equivalent to (elementary) Spin orbits on the unit sphere. When n = 3, this implies that there is a bijection between Um ( R ) E ( R ) and the E ( R )-orbits of 4 × n >
3, this also describes in asimple matrix form, the composition of unimodular rows defined by vander Kallen (using Weak Mennicke symbols). Perhaps more strikingly,with this approach, we now see the possibility of new orbit structures forboth unimodular rows (using octonion multiplication) and for generalquadratic spaces. Introduction
When multiple research areas evolve around the same object, one expectsthat there is a connection between them. The more distinct the methodsare, the more fruitful the connection will be. In this paper, we’ll explore thisdouble life for unimodular rows . Though unimodular rows are primarily usedas a tool to study Projective modules, we’ll see here that they can also befruitfully employed from the perspective of Quadratic forms and Spin groups.One consequence of this approach is that it gives a neat interpretation tosome surprising results like the Vaserstein symbol, through a simple com-position law operating in the background. On the other hand, we arrive atnew questions in this development via quadratic forms. We currently know(through the work of van der Kallen [vdk2] and others) that the Vasersteinsymbol can be generalized to a group law on certain (higher-dimensional)orbit-spaces of unimodular rows. But now, when reinterpreted as a resultin quadratic forms, there is the exciting possibility that such group lawsmay generalize beyond hyperbolic quadratic spaces. In particular, we see(in Part C) that Vaserstein composition corresponds to the special case ofsplit quaternions, and for any other composition algebra, we have a similarcomposition law.
Date : Let R be any commutative ring. Take vectors v, w ∈ R such that v · w ⊺ =1. (Then v is said to be a unimodular row of length 3). Here are a few placeswhere unimodular rows turn up.1.1. First life : Cancellation of Projective modules.
The study of pro-jective modules is one of the primary motivations to investigate unimodularrows. Consider the map R n → R , given by v → v · w ⊺ . When v is a unimodular row, the kernel becomes a projective module.One way to show that a stably-free projective module is free is to showthat the corresponding unimodular row appears as a row in a matrix in SL n ( R ). Thus the interest in unimodular rows began, and grew with theQuillen-Suslin theorem (also known as Serre’s problem) which states thatfinitely generated projective modules over polynomial-rings are free - Quillenreceived a Fields medal in 1978 in part for his proof of the theorem. As onegoes beyond polynomial rings, the orbits may not be trivial, leading to thestudy of quotients such as U m n ( R ) /SL n ( R ) and U m n ( R ) /E n ( R ) and thereis a rich array of results stating conditions under which these orbits have anabelian group structure. As we’ll see, these same orbits can also be examinedfrom a different point of view, as Spin-orbits on the unit sphere.1.2. Second life : Group structures on spheres.
Consider the space H ( R ) = R ⊕ ( R ) ∗ , equipped with a quadratic form q ( x, y ) = x · y ⊺ . Suppose there is another element w ′ ∈ R such that v · w ′ ⊺ = v · w ⊺ = 1.Then it turns out that the two points on the unit sphere - ( v, w ) and ( v, w ′ )- lie on the same orbit under the action of Epin ( R ), the elementary Spingroup (Theorem 4.1).This gives us the map, v → ( v, w )Epin ( R )We’ll see that the kernel of the above map is the orbit of w under the actionof the elementary linear group E ( R ). vE ( R ) ←→ ( v, w )Epin ( R )The seminal paper of Vaserstein-Suslin [SV] introduced the Vasersteinsymbol and contains some hints of the above bijection (though they don’ttalk about Spin groups). In this paper, we will prove this bijection andgeneralize it beyond n = 3 to any n (Theorem 4.4). Let U m n ( R ) denotethe set of unimodular rows of length n and U n − ( R ) be the hyperbolic unitsphere. We will prove that there is a bijection WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 3
U m n ( R ) E n ( R ) ←→ U n − ( R )Epin n ( R ) = U n − ( R ) EO n ( R )In short, orbits of unimodular rows can be studied as orbits of points onthe unit sphere whose geometry is more familiar to us. When n = 3, wehave Epin ( R ) ∼ = E ( R ) which will be used to show that there is a bijectionbetween Um ( R ) E ( R ) and the E ( R )-orbits of 4 × M ∼ gM g T for g ∈ E ( R )), explaining to some extent theVaserstein symbol (Theorem 6.1).The second contribution of this paper is the introduction of a new compo-sition law that holds on certain subspaces of the hyperbolic space H ( R n ) = R n ⊕ ( R n ) ∗ (generalizing Quarternion multiplication). This composition law(on matrices) generalizes the Vaserstein symbol for n ≥ R , whereas it seems necessary to place such restrictions on thebase ring R to get group structures on unimodular rows. What is the rea-son for this dependence on the dimension of R , and how do we arrive at this dependence : in this case, d ≤ n −
3? Looking back, there are tworesults that hint at a general composition law operating in the background- one is the Vaserstein symbol (see Part B), the other being the Mennicke-Newman Lemma (see [vdk2, Lemma 3.2]) that essentially says that underthe above dimension restrictions, one can project two points of the unitsphere U n − ( R ) onto the same ( n + 1)-dimensional subspace, where thecomposition law can operate.This investigation using quadratic forms opens the door for research intwo striking general directions :a. In Section 9, we use the multiplication of split-octonions to definea (nonassociative) composition law on ( n + 4)-dimension subspacesof H ( R n ), suggesting that there may be a quasigroup structure onorbits of unimodular rows.b. Let ( V, q ) be a general quadratic space, and U the set of unit vectorsof V ( q ( x ) = 1). When is there a group structure on the orbit spaces USpin ( V ) ?1.3. The other lives of unimodular rows.
The vector v = ( a, b, c ) alsocorresponds to coefficients of the quadratic form ax + bxy + cy . The condi-tion v · w ⊺ = 1 can then be seen as a restriction to primitive quadratic forms.In the study of unimodular rows, one is mainly concerned with SL ( R ) or-bits of U m ( R ), whereas Gauss’s composition gives a group structure on the SL ( Z ) orbits of binary quadratic forms. It is known that Gauss’s compo-sition extends to an arbitrary base ring (see [K] and [W]). It is also knownthat if ∈ R and the discriminant b − ac is a square, then the unimodularrow ( a, b, c ) is completable and the corresponding projective module is free VINEETH CHINTALA (see [KM] or [Ko]). But it is not known whether there are deeper connec-tions between projective modules and composition laws for quadratic andhigher forms - an intriguing line to pursue, that hopefully future researchcan shed some light on.
Remark 1.4.
There are many other active areas related to unimodularrows - notably, Euler class groups ([BRS, DTZ]), Grothendieck-Witt groups([FRS]), A -homotopy theory ([AF1, AF2]) and Suslin Matrices (see [RJ]for a survey).When R has (Krull) dimension d , J. Fasel has given an interpretation of Um d +1 ( R ) E d +1 ( R ) in terms of cohomology (see [F1]). More recently this quotientspace has been explicitly computed in [DTZ] for some rings. Ravi Rao andSelby Jose have written a series of papers ([JR1, JR2]) examining generalquotients Um n ( R ) E n ( R ) by studying the algebraic properties of Suslin matrices. Asyou can tell, the behaviour of the quotient Um n ( R ) E n ( R ) depends on the base ring R (especially its Krull dimension), and there is a continued trend simplifyingthe hypothesis on the base ring to construct and analyze the structure ofthe orbit spaces (see for example [FRS, GRK, GGR, SS] or Part II of therecent conference proceedings [AHS]).The Vaserstein symbol gives a symplectic structure to the orbit-spacesof unimodular rows and plays an important role in the study of stably-free modules. It was first introduced in [SV, Section 5] where orbits ofunimodular rows were investigated under the action of both linear and sym-plectic groups. Further investigation of the symplectic orbits can be foundin [CR1, CR2, TS2]. The recent work of T. Syed [TS1] generalizes theVasertstein symbol to study the orbit spaces Um ( R + P ) E ( R + P ) , where P is a rank-2projective module with a fixed trivialization of its determinant.A. Asok and J. Fasel have provided an interpretation of the Vasersteinsymbol in terms of A -homotopy theory (see [F2]) and we will explain thisconnection briefly in Part B. In [FRS, Theorem 7.5] the Vaserstein symbolis used to prove that stably free modules of rank d − R is a smooth affine k -algebra of dimension d ≥ k is an algebraically closed field and d − ∈ k ), thus settling a long-standing question of A. Suslin.Perhaps some day, another mathematician will write a “many lives” gen-eralization of this paper. Contents
Part A. The bijection between (elementary) Spin-orbits onthe sphere and the elementary orbits of unimodular rows Part B. Interpreting Vaserstein symbol using Spin groups WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 5
Part C. A general Composition law
Overview.
The paper is broken down into three parts and can be readnon-linearly. A reader whose main interest is unimodular rows may beginwith Part A, where the connection to Spin groups is explored in detail.Alternatively, a person who is curious about general quadratic forms mayfind it profitable to look at Part C first, where a new composition law isdefined using the multiplication in composition algebras. Here the Vasersteincomposition (for unimodular rows) corresponds to the special case of splitquaternions. Finally, those who are comfortable with both the worlds andprefer to quickly know what is going on, may begin with Part B whichacts as a bridge (examining the Vaserstein symbol), and then read aroundaccordingly.Essentially the paper makes two contributions. First, we look at (elemen-tary) orbits of unimodular rows and prove that they correspond to (elemen-tary) Spin-orbits on the unit sphere. Secondly, we introduce a compositionlaw - that holds in certain ( n + 2)-dimension subspaces of H ( R n ). Thiscomposition (in terms of matrices) follows a simple recursive rule, startingwith the multiplication of split-quaternions. When n = 3, it has the same properties as the composition law (on unimodular rows) introduced by L.Vaserstein. For general n , it describes in a simple matrix form, the compo-sition defined by van der Kallen’s using weak mennicke symbols. This laysthe foundation for the study of unimodular rows using Spin groups. Thegeneral formulation of the composition law also raises the possibility of neworbit structures using octonion multiplication.1.6. Notation.
All modules in the paper are free R -modules over somecommutative ring R . The results proved in the paper hold for all commu-tative rings. Part A. The bijection between (elementary)Spin-orbits on the sphere and theelementary orbits of unimodular rows Preliminaries : From Clifford algebra to Suslin matrices
Clifford Algebras.
Let V be a free R -module where R is any com-mutative ring. If we equip V with a quadratic form q , then ( V, q ) is calleda quadratic space. The algebra Cl(
V, q ) is the “freest” algebra generated by
VINEETH CHINTALA V subject to the condition x = q ( x ) for all x ∈ V . More precisely, Cl( V, q )is the quotient of the tensor algebra T ( V ) = R ⊕ V ⊕ V ⊗ ⊕ · · · ⊕ V ⊗ n ⊕ · · · by the two sided ideal I ( V, q ) generated by all the elements x ⊗ x − q ( x ) with x ∈ V .For the purpose of this article, we only need to know two basic propertiesof Clifford algebras : • Z -grading : Grading T ( V ) by even and odd degrees, it follows thatthe Clifford algebra has a Z -grading Cl( V, q ) = Cl ⊕ Cl such that V ⊆ Cl and Cl i Cl j ⊆ Cl i + j ( i, j mod 2). • Universal property : Given any associative algebra A over R and anylinear map j : V → A such that j ( x ) = q ( x ) for all x ∈ V , there is a unique R -algebra homomorphism f : Cl( V, q ) → A suchthat f ◦ i = j .Let Cl denote the Clifford algebra of the quadratic space H ( R n ) := R n ⊕ R n ∗ , with q ( v, w ) = v · w ⊺ . We’ll now give an explicit representation of Cl ∼ = M n ( R ) using what are called Suslin matrices. For a detailed exposition,see [CV1].2.2. Suslin matrices.
For any two vectors v = ( a , · · · , a n ) and w =( b , · · · , b n ) in R n , the Suslin matrix S ( v, w ) is defined as follows :For n = 2, define S ( v, w ) = (cid:0) a a − b b (cid:1) S ( v, w ) = (cid:16) b − a b a (cid:17) For the general case, write v = ( a , v ′ ) and w = ( b , w ′ ) with v ′ , w ′ ∈ R n − .Then S ( v, w ) = (cid:20) a S ( v ′ ,w ′ ) − S ( v ′ ,w ′ ) b (cid:21) , S ( v, w ) = (cid:20) b − S ( v ′ ,w ′ ) S ( v ′ ,w ′ ) a (cid:21) The matrix S = S ( v, w ) has size 2 n − × n − and has the following prop-erties :a. S ( v, w ) = S ( w, v ) ⊺ .b. SS = SS = ( v · w ⊺ ) I n − . In his paper [S], A. Suslin then describes a sequence of matrices J n ∈ M n ( R )by the recurrence formula WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 7 J n = n = 0 (cid:16) J n − − J n − (cid:17) for n even (cid:16) J n − − J n − (cid:17) for n odd . One can check by induction that
J J ⊺ = 1. Importantly, their relation toClifford algebras comes from the following equations : J n − S ⊺ n − J ⊺ n − = S n − for n odd, S n − for n even. (1)As J J ⊺ = 1, it follows that M ∗ = J M ⊺ J ⊺ is an involution of M n ( R ).The map φ : H ( R n ) → M n ( R ) defined by φ ( v, w ) = (cid:0) S n − ( v,w ) S n − ( v,w ) 0 (cid:1) induces an R -algebra homomorphism φ : Cl → M n ( R ). In fact φ is an iso-morphism (Section 3.1, [CV1]); the elements φ ( v, w ) give a set of generatorsof the Clifford algebra. In addition, the involution M ∗ = J M ⊺ J ⊺ turns outbe what is called the standard involution of the Clifford algebra (Theorem4.1, [CV1]). Note that the quadratic form is q ( v, w ) = S ( v, w ) S ( v, w ). For S i = S ( v i , w i ), the corresponding bilinear form is h S , S i = S S + S S = v · w ⊺ + v · w ⊺ . Properties of the basis vectors.
Let E i = S n − ( e i , , F i = S n − (0 , f i ) . Notice that E = (cid:0) (cid:1) and F = (cid:0) (cid:1) . For i > E i , F i are ofthe form (cid:0) X − X (cid:1) for some Suslin matrix X with XX = 0.It is easy to check that the elements E i , F i satisfy the following elementaryproperties. Lemma 2.4.
Let X k ∈ { E k , F k } for ≤ k ≤ n . Let i = 1 . Thena. X = X and X + X = 1 b. X i = − X i and X i = 0 .c. X i X = X X i . Theorem 2.5.
Let X k ∈ { E k , F k } for ≤ k ≤ n . We have the followingcommutator relations whenever / ∈ { i, j } : λ X i X j = [1 + λ X i X , X X j ] VINEETH CHINTALA
Proof.
It follows from Lemma 2 . X i X is 1 − X i X .Moreover, since X i = X j = 0 and X i X j + X j X i = h X i , X j i = 0, any termwhere X i or X j appears twice is zero. Thus we are left with[1 + λ X i X , X X j ] = 1 − λ X X j X i X + λ X i X X j = 1 + λ X i X j ( X + X )Since X + X = 1 we are done. (cid:3) The Elementary Spin Group
As stated earlier, the Clifford algebra is a Z -graded algebra Cl = Cl ⊕ Cl . Under the isomorphism φ : Cl ∼ = M n ( R ), the elements of Cl corre-spond to matrices of the form (cid:0) g g (cid:1) .The Spin group is defined asSpin n ( R ) := { x ∈ Cl | xx ∗ = 1 and xH ( R n ) x − = H ( R n ) } . Just like we have the elementary group E n ( R ) corresponding to SL n ( R ), wehave similar analogues for the orthogonal and Spin groups. Definition 3.1. a. Let e ij denote the matrix with in the ( i, j ) position and zeroeseverywhere else. For i = j , define E ij ( λ ) = 1 + λe ij The matrices E ij ( λ ) are called elementary matrices and the groupgenerated by n × n elementary matrices is called the elementary group E n ( R ) .b. Let ∂ denote the permutation (1 n + 1) ... ( n n ) . We define for ≤ i = j ≤ n , λ ∈ R , E oij ( λ ) = I n + λ ( e ij − e ∂ ( j ) ∂ ( i ) ) . We call these the elementary orthogonal matrices and the group gen-erated by them is called the elementary orthogonal group EO n ( R ) . c. From the definition of the Spin group, we have the map π : Spin n ( R ) → O n ( R ) given by π ( g ) : ( v, w ) → g · ( v, w ) · g − for g ∈ Spin n ( R ) . We denote by
Epin n ( R ) the inverse image of EO n ( R ) under π . The group Epin n ( R ) satisfies the following exact sequence (see [B2, p. 189])1 → µ ( R ) → Epin n ( R ) → EO n ( R ) → µ ( R ) = { x ∈ R : x = 1 } . WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 9
Since π : Epin n ( R ) → EO n ( R ) is surjective, it follows that U n − ( R )Epin n ( R ) = U n − ( R ) EO n ( R ) , (2)where U n − ( R ) is the unit sphere in H ( R n ). Lemma 3.2.
There is a homomorphism H : E n ( R ) → EO n ( R ) given by ε → (cid:16) ε ε ⊺ − (cid:17) ∈ EO n ( R ) .Proof. The lemma follows from the observation that H ( E ij ( λ )) = E oij ( λ ) . (cid:3) Generators of
Epin n ( R ) . Let V = R n with standard basis e , · · · , e n and dual basis f , · · · , f n for V ∗ . We will identify H ( V ) with the corresponding matrices in the Cliffordalgebra. In terms of Suslin matrices, e i = h S n − ( e i , S n − ( e i ,
0) 0 i , f i = h S n − (0 ,f i ) S n − (0 ,f i ) 0 i It can be proved (see [B2, Section 4.3]) that Epin n ( R ) is generated byelements of the form 1 + λe i e j , λe i f j , λf i f j with λ ∈ R , 1 ≤ i, j ≤ n , i = j .Let ( x k , X k ) ∈ { ( e k , E k ) , ( f k , F k ) } . Then the generator 1 + λx i x j corre-sponds to the matrix φ (1 + λx i x j ) = h λ X i X j
00 1+ λ X i X j i Since e i , e are orthogonal we have e i e = − e e i . Similarly f i f = − f f i .By also taking into account the commutator relations in Theorem 2.5, wefind that Epin n ( R ) is generated by the (smaller) set of elements of the type1 + λe e i , λe f i , λf e i , λf f i . The action of the Epin group.
So how do the above generators acton the quadratic space?Suppose g = h λ E E i
00 1+ λ ¯ E ¯ E i i . Since E i = − E i and E E i = E i E , we have g h S ( v,w ) S ( v,w ) 0 i g − = h S ( v ′ ,w ′ ) S ( v ′ ,w ′ ) 0 i , where S ( v ′ , w ′ ) = (1 + λ E E i ) · S ( v, w ) · (1 − λ ¯ E ¯ E i )= (1 + λ E E i ) · S ( v, w ) · (1 + λ E i E )Recall that for i > E i , F i are of the form (cid:0) X − X (cid:1) for someSuslin matrix X with XX = 0. Then 1 + λ E E i and 1 + λ E i E will be equalto (cid:0) λ X (cid:1) and (cid:0) − λ X (cid:1) respectively. Lemma 3.5.
Let X , T ∈ M k ( R ) be two Suslin matrices and XX = 0 . Let S = (cid:16) a T − T b (cid:17) . Then (cid:0) X (cid:1) S (cid:0) − X (cid:1) = (cid:20) a −h X , T i T + b X − T − b X b (cid:21) , (cid:0) X (cid:1) S (cid:0) − X (cid:1) = (cid:20) a T − a X − T + a X b + h X , T i (cid:21) . Proof.
Note that h X , T i = XT + TX . The proof follows by straightforwardmatrix multiplication. (cid:3) Lemma 3.6.
Let X ∈ { λ E i , λ F i } where i = 1 and λ ∈ R . Suppose v · w ⊺ = 1 for two vectors v, w ∈ R n +1 . Then (cid:0) X (cid:1) S ( v, w ) (cid:0) − X (cid:1) = S ( vε, wε ⊺ − ) (cid:0) X (cid:1) S ( v, w ) (cid:0) − X (cid:1) = S ( vσ, wσ ⊺ − ) for some ε, σ ∈ E n +1 ( R ) .Proof. Let X = λ E i . The proof is similar in the other case. Write v =( a , · · · , a n ) and w = ( b , · · · , b n ).From Lemma 3.5, we have (cid:0) λ E i (cid:1) S ( v, w ) (cid:0) − λ E i (cid:1) = S ( v ′ , w ′ ), where (cid:20) v ′ w ′ (cid:21) = (cid:20) ( a − λb i , · · · , a i + λb, · · · , a n ) w (cid:21) . Since v ′ · w ⊺ = v · w ⊺ = 1, it follows from [S, Corollary 2.7] that the matrices ε = I n + w ⊺ ( v − v ′ ) , ( ε ⊺ ) − = I n − ( v − v ′ ) ⊺ w are in E n ( R ). We have ( vε, wε ⊺ − ) = ( v ′ , w ) . For the second part, taking X = λ E i in Lemma 3.5, we have (cid:2) X (cid:3) S ( v, w ) (cid:2) − X (cid:3) = S ( v ′′ , w ′′ ) , where (cid:20) v ′′ w ′′ (cid:21) = (cid:20) ( a , · · · , a i − λa , · · · , a n )( b + λb i , · · · , b n ) (cid:21) . Clearly ( v ′′ , w ′′ ) = ( vσ, wσ ⊺ − ) where σ = E i ( − λ ). (cid:3) The bijection between
Epin n ( R ) and E n ( R ) orbits We are now ready to prove the bijection between E n ( R )-orbits of unimod-ular rows and Epin n ( R )-orbits on the unit sphere in H ( R n ) = R n ⊕ R n ∗ .We’ll break it down into simple parts with each part explaining one aspectof the bijection. WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 11
Theorem 4.1.
Let q ( v, w ) = 1 and g ∈ Epin n ( R ) . Then g ( v, w ) g − = ( vσ, wσ ⊺ − ) for some σ ∈ E n ( R ) .Proof. It is enough to prove the theorem for the generators of the Epin n ( R )group. Let g = (cid:0) g g (cid:1) be one of the generators1 + λe e i , λe f i , λf e i , λf f i . Then g determines g , and g is either of the form (cid:0) X (cid:1) or (cid:0) − X (cid:1) with X ∈ { λ E i − , λ F i − } . The rest follows from Lemma 3.6. (cid:3) Remark 4.2.
There are two papers in the literature which prove somevariation of the above theorem, though neither of them discuss Spin groups.The special case n = 3 was considered in the proof of Coroallary 7.4, [SV],and an alternate approach can be found in Lemma 3.2 of [JR1]. Both thepapers study different group structures and connect them to the elementary-group actions on unimodular rows. We will interpret the Vaserstein symbolusing Spin groups in Part B of the paper. Theorem 4.3.
Let n ≥ . If q ( v, w ) = q ( v, w ) = 1 , then ( v, w ) and ( v, w ) are in the same EO n ( R ) and Epin n ( R ) orbits.Proof. By our hypothesis, we have v · w ⊺ = v · w ⊺ = 1. Then it follows, from[S, Corollary 2.7], that the matrix ε := I n + v ⊺ ( w − w ) ∈ E n ( R ) . Since w · ε = w , both w , w lie in the same E n ( R ) orbit.By Lemma 3.2, we have H : ε → (cid:16) ε ⊺ − ε (cid:17) ∈ EO n ( R ). Since ε ⊺ − = I n − ( w − w ) ⊺ v , it is easy to check that w ε = w ,vε ⊺ − = v. Therefore ( v, w ) and ( v, w ) lie in same EO n ( R ) orbit, and so by Equation2 they lie in the same Epin n ( R ) orbit. (cid:3) Let U n − ( R ) be the unit sphere in H ( R n ). By the above theorem, themap U m n ( R ) → U n − ( R ) Epin n ( R ) given by v → ( v, w ) is well defined. Theorem 4.4.
Let ( v , w ) , ( v , w ) be two points on the unit sphere U n − ( R ) ,where n ≥ . Then ( v , w ) ∼ Epin2 n ( R ) ( v , w ) if and only if v ∼ En ( R ) v .In other words, there is a bijection between the sets (of orbits) U m n ( R ) E n ( R ) ←→ U n − ( R )Epin n ( R ) = U n − ( R ) EO n ( R ) .2 VINEETH CHINTALA
Let ( v , w ) , ( v , w ) be two points on the unit sphere U n − ( R ) ,where n ≥ . Then ( v , w ) ∼ Epin2 n ( R ) ( v , w ) if and only if v ∼ En ( R ) v .In other words, there is a bijection between the sets (of orbits) U m n ( R ) E n ( R ) ←→ U n − ( R )Epin n ( R ) = U n − ( R ) EO n ( R ) .2 VINEETH CHINTALA Proof.
Suppose for two unimodular rows v , v , we have v · ε = v for some ε ∈ E n ( R ). Then Theorem 4 . v , w ) ∼ Epin2 n ( R ) ( v · ε, w · ε ⊺ − ) ∼ Epin2 n ( R ) ( v , w )On the other hand, suppose ( v , w ) ∼ Epin2 n ( R ) ( v , w ). Then Theorem 4 . v ∼ En ( R ) v . Therefore we have a bijection Um n ( R ) E n ( R ) ←→ U n − ( R )Epin n ( R ) . (cid:3) Corollary 4.5.
Let ( v , w ) , ( v , w ) be two points on the unit sphere U n − ( R ) ,where n ≥ . Then ( v , w ) ∼ EO n ( R ) ( v , w ) if and only if w ∼ En ( R ) w . The above bijection says that for any g ∈ Epin n ( R ) and a point ( v, w )on the unit sphere, g ( v, w ) g − = ( vσ, wσ ⊺ − )for some σ ∈ E n ( R ). Here, the element σ ∈ E n ( R ) may vary with the choiceof ( v, w ). It should be stressed that the above bijection does not imply thatthe groups E n ( R ) and Epin n ( R ) are isomorphic. Only the correspondingorbits spaces are in bijection. Part B. Interpreting Vaserstein symbol usingSpin groups
In this part, we’ll return to the case n = 3 and examine the Vasersteinsymbol. 5. The Vaserstein symbol
Definition 5.1. ( [SV, p. 945] ) The elementary symplectic-Witt group W E ( R ) is an abelian group consisting of (equivalent classes of ) skew-symmetric ma-trices. For skew-symmetric matrices α r ∈ M r ( R ) their sum is defined as α r ⊥ α s := (cid:0) α r α s (cid:1) ∈ M r + s ( R ) . The identity element is ψ r = ψ r − ⊥ ψ where ψ = (cid:0) − (cid:1) . Two matrices α r , α s are said to be equivalent if α r ⊥ ψ s + l = ε ( α s ⊥ ψ r + l ) ε ⊺ , for some l ≥ and ε ∈ E ( R ) . The Vaserstein symbol is a map Um ( R ) E ( R ) → W E ( R ), giving a symplecticstructure on orbits of unimodular rows. This is done by identifying a pointon the unit sphere ( v, w ) ∈ H ( R ) with a 4 × WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 13
Let v = ( a , a , a ) and w = ( b , b , b ). Recall from Section 2.2 that S ( v, w ) = a a a − b b − b a b − b − a b ! , J = (cid:18) − −
10 0 1 0 (cid:19) , and J S ⊺ J ⊺ = S (from Equation 1). Since J − = J ⊺ = − J , this can berewritten as ( S J ) ⊺ = − S J. Define V ( v, w ) := S ( v, w ) J = a a − a − a b b − a − b − b a − b b . The matrix V ( v, w ) is skew-symmetric and represents an element of W E ( R ).In the next section we’ll break down Vaserstein symbol into two parts andinterpret it using Spin groups :a. Let A ( R ) denote the set of 4 × Um ( R ) E ( R ) ↔ A ( R ) E ( R ) . Aswe’ll see, this follows from the isomorphism Epin R ∼ = E ( R ) andthen utilizing the results from Part A to get U m ( R ) E ( R ) ↔ U ( R )Epin ( R ) ↔ A ( R ) E ( R ) . b. Then the obvious inclusion map gives us A ( R ) → W E ( R ), thusrevealing the Witt-group structure on orbits of unimodular rows. Remark 5.2.
The Vaserstein symbol was introduced in [SV, Section 5] tostudy orbits of unimodular rows. Suslin and Vaserstein studied the injectiv-ity and surjectivity of the Vaserstein symbol and proved that it is a bijectionif dim ( R ) ≤ A -homotopy theory. The paper [F2] explainsthis connection in detail (also see [AF2, Theorem 4.3.1]). Following [F2],let k be a perfect field and Q be the smooth affine quadric with k [ Q ] = k [ x , x , x , y , y , y ] / h P x i y i = 1 i . For any smooth affine k -scheme X = Spec ( R ), there is a natural bijection [ X, Q ] A = [ X, A \ A = U m ( R ) /E ( R ).Moreover Q is isomorphic to the quotient of algebraic varieties SL /Sp giving us the composite map Q → SL /Sp → SL/Sp . It turns outthat the quotient
SL/Sp represents the (reduced) higher Grothendieck-Wittgroup GW ( X ) which coincides with W E ( R ) for any smooth affine variety X = Spec ( R ). Thus one has the following interpretation of the Vasersteinsymbol U m ( R ) /E ( R ) = [ X, Q ] A → [ X, SL/Sp ] A = W E ( R ) . The dictionary between Vaserstein and Suslin matrices
We’ll borrow results from [CV1] on the connection between Suslin matri-ces and Clifford algebras. Specifically we need the well-known exceptionalisomorphisms Spin ( R ) ∼ = SL ( R ) and Epin ( R ) ∼ = E ( R ). (For a proofusing Suslin matrices, see [CV1, Theorems 7.1, 8.4]).Define ∗ to be the involution on M ( R ) given by M ∗ = J M ⊺ J ⊺ where J = (cid:18) − −
10 0 1 0 (cid:19) . Note that ∗ is an involution because J ⊺ = − J = J − .Let’s identify the Suslin matrix S ( v, w ) with the element ( v, w ) in thequadratic space H ( R ). Under the isomorphism ψ : Spin ( R ) ∼ = SL ( R ),the Spin group behaves as follows : for g ∈ SL ( R ), the action is given by g • S = gSg ∗ . Simplifying the notation, we’ll sometimes write S , V insteadof S ( v, w ) , V ( v, w ).Any 4 × V ( v, w ), corresponding tothe element ( v, w ) ∈ H ( R ). Let A ( R ) denote the set of all such matriceswith v · w ⊺ = 1 (the unit sphere in H ( R )). The group SL ( R ) acts on thematrices V ( v, w ) as ( g, V ) → gV g T . Recall that the unit sphere in H ( R )is denoted by U ( R ). Theorem 6.1.
We have the bijection U ( R )Spin ( R ) ↔ A ( R ) SL ( R ) . Therefore,
U m ( R ) E ( R ) ↔ U ( R ) EO ( R ) = U ( R )Epin ( R ) ↔ A ( R ) E ( R ) where v → S ( v, w ) → V ( v, w ) for any element ( v, w ) ∈ U ( R ) .Proof. The bijection between the E ( R )-orbits of unimodular rows and Epin ( R )-orbits on the unit sphere follows from Theorem 4 . A ( R ) corresponds to the unit sphere in H ( R ). We’llnow show that the group actions on H ( R ) are the same. Remember that V = S J , or equivalently S = − V J . Since J ⊺ = − J , it follows that g • S = g S g ∗ = g S J g ⊺ J ⊺ = − ( gV g ⊺ ) J. In other words, for any g ∈ SL ( R ), if g • S ( v, w ) = S ( v ′ , w ′ ) then gV ( v, w ) g ⊺ = V ( v ′ , w ′ ). This means that the SL ( R ) (and E ( R )) action on 4 × M → gM g ⊺ is the same as the Spin ( R ) (and Epin ( R ))action on the quadratic space H ( R ). Restricted to the unit sphere in H ( R ), the bijection is clear. (cid:3) The above correspondence gives another proof of the following well-knownexceptional isomorphism.
Theorem 6.2.
Spin ( R ) ∼ = Sp ( R ) . WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 15
Proof.
The proof follows by identifying Spin ( R ) as a subgroup of Spin ( R )which fixes ( v, w ) = (1 , , , , , ( R ) then corre-spond to matrices g ∈ SL ( R ) such that gg ∗ = 1. In other words, gJ g ⊺ = J ,which is precisely the group Sp ( R ). (cid:3) The Vaserstein symbol V : Um ( R ) E ( R ) → W E ( R ) can thus be decomposed as V : U m ( R ) E ( R ) ∼ = A ( R ) E ( R ) → W E ( R ) . The injectivity (surjectivity) of the Vasersetin symbol boils down to theinjectivity (surjectivity) of the map A ( R ) E ( R ) → W E ( R ), which is defined nat-urally via the inclusion map. The interpretation in terms of Spin groups issummarized in the table below :Vaserstein symbol The Spin group interpretation4 × V ( v, w ) 4 × S ( v, w ), ( V = S J )Action of E ( R ) : ( g, V ) → gV g ⊺ Action of Epin ( R ) : g • S = g S g ∗ A ( R ) U ( R )Orbits of Unimodular rows : A ( R ) E ( R ) Orbits on the sphere ( v · w ⊺ = 1) : U ( R )Epin ( R ) SL ( R ) , E ( R ) Spin ( R ) , Epin ( R ) Sp ( R ) Spin ( R )6.3. A question about K Spin ( R ) . One also has the map U n − ( R )Epin n ( R ) → Spin n R Epin n ( R ) → K Spin ( R ). What is the relation between W E ( R ) and theabelian group K Spin ( R )?6.4. Vaserstein composition.
The paper [SV] also introduced a composi-tion law on unimodular rows ([SV, Theorem 5.2]). The composition law waslater generalized to
U m n ( R ) by W. van der Kallen using Weak Mennickesymbols as follows (see [vdk2, Lemma 3.4]) :Let v = ( a , a , a , · · · , a n ) and v = ( c , c , a , · · · , a n ) be two unimod-ular rows and choose d , d such that the determinant of β = (cid:0) c c − d d (cid:1) hasimage 1 in R/ h a , · · · , a n i . Then wms ( v ) wms ( v ) = wms ( p, q, a , · · · a n )where ( p, q ) = ( a , a ) β .In Part C, we’ll introduce a new composition law on certain subspacesof H ( R n ) satisfying the same properties. Moreover this law has the nice feature that it is expressed recursively using matrices. It turns out that thiscomposition of unimodular rows is a special case of a more general law, whichacts on certain subspaces of A ⊕ H ( R n ) where A is a composition algebra.The Vaserstein composition corresponds to the case where the compositionalgebra is the algebra of split quaternions.As an illustration of the results in Part C, we’ll now interpret Vaserstein’scomposition rule using Suslin matrices for the case n = 3. Let ( v , w ) and( v , w ) be two points on the unit sphere of H ( R ).Let S i = S ( v i , w i ). We have S = ( a α − α b ) and S = (cid:16) a β − β b ′ (cid:17) , where α = (cid:0) a a − b b (cid:1) and β = (cid:0) c c − d d (cid:1) are 2 × S ⊙ S := a αβ − αβ b + b ′ − abb ′ The element S ⊙ S is also a Suslin matrix and q ( S ⊙ S ) = q ( S ) q ( S ) . Moreover the composition S ( v ′ , w ′ ) = S ⊙ S is similar to the Vasersteinsymbol, where the product of the matrices α, β gives us the values of theunimodular row w ′ . Specifically, we have v ′ = ( a, ( a , a ) β ). Part C. A general Composition law Starting with the multiplication of Composition algebras
Composition algebras.
A composition algebra (
A, q ) over R is a (notnecessarily associative) R -algebra, equipped with a (non-degenerate) qua-dratic form satisfying q ( xy ) = q ( x ) q ( y ) for all x, y ∈ A . We’ll assume A isa free R -module. It is known that rank( A ) has to be 1, 2, 4, or 8 (see [Kn,V. 7.1.6]). For any composition algebra ( A, q ), there is an involution α → α such that q ( α ) = α ¯ α , for all α ∈ A .The following construction is inspired by the construction of Suslin ma-trices. Let ( A, q ) be any composition algebra. Consider the quadraticspace A ⊕ H ( R ), where H ( R ) is a hyperbolic plane. For each element( α, a, b ) ∈ A ⊕ H ( R ), the quadratic form is given by q ( α, a, b ) = α ¯ α + ab. One can represent ( α, a, b ) as a matrix Z = ( a α − α b ). Define Z = (cid:0) b − αα a (cid:1) .Then q ( Z ) = ZZ = ZZ = α ¯ α + ab. For any such matrix, we’ll sometimes write q Z instead of q ( Z ). One ofthe reasons we rewrite the elements as 2 × WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 17 to express the composition law and generalize the analysis to A ⊕ H ( R n ).In addition, as we shall see later, this matrix representation gives a simpledescription of the Clifford algebra and the corresponding Spin groups.7.2. Composition law for hyperplanes of A ⊕ H ( R ) . Let X = (cid:18) a α − α b (cid:19) and Y = (cid:18) a β − β b ′ (cid:19) . When q X = q Y = 1, define X ⊙ Y := a αβ − αβ b + b ′ − abb ′ Then q ( X ⊙ Y ) = 1.For general X, Y define X ⊙ Y := a αβ − αβ bq Y + b ′ q X − abb ′ . From the equations α ¯ α = q X − ab, β ¯ β = q Y − ab ′ , it follows that q ( X ⊙ Y ) = q ( X ) q ( Y ) . When the underlying composition algebra is associative, the operation ⊙ isalso associative with the identity element (cid:16) a − (cid:17) .If we take a = b = b ′ = 0, then X ⊙ Y = (cid:18) αβ − αβ (cid:19) corresponds to themultiplication in the composition algebra. When A ∼ = M ( R ) is the algebraof split quaternions, then the above composition law gives us the Vasersteincomposition on unimodular rows stated in Part B.7.3. The quadratic space A ⊕ H ( R n ) . Consider next the quadratic space A ⊕ H ( R n ), where H ( R n ) = R n ⊕ R n ∗ .For each element ( α, v, w ) ∈ A ⊕ H ( R n ), the quadratic form is given by q ( α, v, w ) = α ¯ α + v · w ⊺ . Here α is an element of the composition algebra A and v, w ∈ R n .By fixing a basis of R n , let us write v = ( a , · · · , a n ) and w = ( b , · · · , b n ).Let Z ( α, v, w ) = (cid:0) a α − α b (cid:1) and Z ( α, v, w ) = (cid:16) b − αα a (cid:17) .For i >
1, define recursively the matrices Z i ( α, v, w ) := (cid:16) a i Z i − − Z i − b i (cid:17) and Z i ( α, v, w ) := (cid:16) b i − Z i − Z i − a i (cid:17) . Then Z i is a 2 i × i matrix and q ( Z i ) = Z i ¯ Z i = ¯ Z i Z i = α ¯ α + a b + · · · + a i b i . Composition law for certain subspaces of A ⊕ H ( R n ) . Fix v = ( a , · · · , a n ) ∈ R n .Let α, β ∈ A, w = ( b , · · · , b n ) and w ′ = ( b ′ , · · · , b ′ n ).Write X i = Z i ( α, v, w ) and Y i = Z i ( β, v, w ′ ). By definition, we have q X i = a i b i + q X i − and q Y i = a i b ′ i + q Y i − . Define the composition X i ⊙ Y i recursively as X i ⊙ Y i := a i X i − ⊙ Y i − − X i − ⊙ Y i − b i q Y i + b ′ i q X i − a i b i b ′ i . By induction, it follows that q X n ⊙ Y n = q X n q Y n . Remark 7.5.
When A ∼ = M ( R ) is the algebra of split quaternions, thematrices Z ( α, v, w ) are Suslin matrices. Let v = ( a , a , a , · · · , a n ) and v = ( c , c , a , · · · , a n ) be two unimodular rows such that v i · w ⊺ i = 1. We’llnow interpret (van der Kallen’s) composition of unimodular rows (which wasdefined in terms of weak mennicke symbols) using ⊙ .Suppose S ( v , w ) ⊙ S ( v , w ) = S ( v , w ) . Then v = ( p, q, a , · · · , a n )where ( p, q ) = ( a , a ) β . Here β = (cid:0) c c − d d (cid:1) where w = ( d , d , · · · , d n ).Clearly the determinant of β has image 1 in R/ h a , · · · , a n i as v · w ⊺ = 1.Therefore wms ( v ) wms ( v ) = wms ( p, q, a , · · · a n ) = wms ( v ) . The Clifford algebra of A ⊕ H ( R n ) : the quaternion case Here we’ll consider the case when A is a quaternion algebra over R . Let V = A ⊕ H ( R n ) and we’ll continue representing its elements ( v, w, α ) as amatrix Z n ( v, w, α ). Notice that Z n ( v, w, α ) ∈ M n ( A ) . Consider the map φ : V → M n +1 ( A ) given by( v, w, α ) → (cid:20) Z n ( v, w, α ) Z n ( v, w, α ) 0 (cid:21) Since φ ( v, w, α ) = q ( v, w, α ), by the universal property of Clifford algebrasthe map lifts to an R -algebra homomorphism φ : Cl ( V ) → M n +1 ( A ) . WO LIVES : COMPOSITIONS OF UNIMODULAR ROWS 19
This is in fact a graded homomorphism, where the even and odd elementsof M n +1 ( A ) correspond to matrices of the form ( ∗ ∗ ) and ( ∗∗ ). Theorem 8.1.
The map φ : Cl ( V ) → M n +1 ( A ) is an isomorphism.Proof. Let ker ( φ ) denote the kernel of φ . Since φ restricts to an injectivemap on V , we have ker ( φ ) ∩ R = { } . Then it follows from [CV2, Theorem2.7] that φ is injective.Since M n +1 ( A ) = M n +1 ( R ) ⊗ A , one can see that its rank is 2 n +4 , thesame as rank( Cl ) = 2 rank( V ) . By dimension arguments, it follows that themap φ is an isomorphism. (cid:3) Remark 8.2.
The paper [CV2] analyzes such embeddings for general qua-dratic spaces, in particular describing the structure of the Clifford algebraand Spin groups.9.
Clifford algebra: the octonion case
The embedding in the endomorphism ring.
Let O be an octonionalgebra. The problem here is that the matrix algebra M n +1 ( O ) is notassociative anymore. However the octonion algebra O has the interestingproperty that α ( αβ ) = q ( α ) β for all α, β ∈ O . (See [Kn, Ch. V, § L : O → End(O) where L α is left-multiplication by α . These maps satisfy the prop-erty that L α L ¯ α = L q ( α ) .We’ll modify the matrices Z i ( v, w, α ) by replacing α with L α in the ma-trix.Define Z ′ ( α, v, w ) = (cid:16) a L α − L α b (cid:17) and Z ′ ( α, v, w ) = (cid:16) b − L α L α a (cid:17) .For i >
1, define recursively the matrices Z ′ i ( α, v, w ) := (cid:18) a i Z ′ i − − Z ′ i − b i (cid:19) and Z ′ i ( α, v, w ) := (cid:18) b i − Z ′ i − Z ′ i − a i (cid:19) .9.2. The Clifford algebra.
We have the map φ : Cl ( V ) → M n +1 (End(O))given by ( v, w, α ) → (cid:20) Z ′ n ( v, w, α ) Z ′ n ( v, w, α ) 0 (cid:21) Theorem 9.3.
The map φ : Cl ( V ) → M n +1 (End(O)) is an isomorphism.Proof. The proof is similar to Theorem 8 .
1. Since φ restricts to an injectivemap on V , we have ker ( φ ) ∩ R = { } . Then it follows from [CV2, Theorem2.7] that φ is injective.Note that rank[End(O)] = 64 because rank( O ) = 8. Since M n +1 (End(O)) = M n +1 ( R ) ⊗ End(O), one can see that its rank is 2 n +8 which is the sameas rank( Cl ) = 2 rank( V ) . By dimension arguments, the map φ is an isomor-phism. (cid:3) Composition in H ( R ) using Octonion multiplication. Let v =( a, v ) and w = ( b, w ), where ( v , w ) ∈ H ( R ). Let us identify elements of H ( R ) with the elements of the split octonion algebra - write O = ( v , w )with q ( O ) = O O = v · w ⊺ . Let X = (cid:16) a O − O b (cid:17) and Y = (cid:16) a O − O b ′ (cid:17) .When q X = q Y = 1, we have X ⊙ Y = (cid:16) a O O − O O b + b ′ − abb ′ (cid:17) The product X ⊙ Y corresponds to another pair ( v ′ , w ′ ) ∈ H ( R ) and q ( X ⊙ Y ) = v ′ · w ′ ⊺ = 1. This composition is obviously non-associative. Acknowledgements.
The author is currently supported by the DST-Inspire Fellowship in India. I would like to thank Ravi Rao, Jean Faseland Anand Sawant for all the conversations and constant encouragement.Special thanks to the Euler International Mathematical Institute, Russiafor their generous invitation to the Algebraic groups conference in 2019,and for their excellent hospitality in St. Petersburg - where some of theearly thoughts in the paper took a clear shape.
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