aa r X i v : . [ m a t h . R A ] F e b AUTOMORPHISMS OF QUANTUM POLYNOMIALS
ASHISH GUPTA
Abstract.
An important step in the determination of the automorphism group of the quantumtorus of rank n (or twisted group algebra of Z n ) is the determination of its so-called non-scalarautomorphisms. We present a new algorithimic approach towards this problem based on thebivector representation V : GL( n, Z ) → GL( (cid:0) n (cid:1) , Z ) of GL( n, Z ) and thus compute the non-scalar automorphism group Aut( Z n , λ ) in several new cases. As an application of our ideas weshow that the quantum polynomial algebra (multiparameter quantum affine space of rank n )has only scalar (or toric) automorphisms provided that the torsion-free rank of the subgroupgenerated by the defining multiparameters is no less than (cid:0) n − (cid:1) + 1 thus improving an earlierresult. We also investigate the question: when is a multiparameter quantum affine space free ofso-called linear automorphisms other than those arising from the action of the n -torus ( F ∗ ) n . Contents
1. Introduction 12. Automorphisms of quantum tori 43. Automorphisms of quantum affine spaces - preliminary facts 94. Automorphisms of quantum affine spaces–Main results 114.1. Examples 13References 151.
Introduction
Quantum polynomials are non-commutative versions of polynomial and Laurent polynomialalgebras. The difference with the ordinary polynomials lies in the fact that the commutativity ofthe variables is replaced by quasi-commutativity, that is, X i X j = q ij X j X i for non-zero scalars q ij .This relation is the Weyl form of the cannonical commutation relation of quantum mechanics. Forthe Laurent case (where each variable X j has an inverse) there other terms in use, for example,quantum torus, twisted group algebra, McConnell–Pettit algebra etc.Quantum polynomials play a key role in the theory of quantum groups [Ar] and also in non-commutative geometry [M]. Their Laurent versions arise in Lie theory as coordinate structures ofextended affine Lie algebras [Ne] and also in the representation theory of nilpotent groups [Br].The question of automorphisms of polynomial algebras is a still open one (the Jacobian conjecture)and it is therefore of interest to investigate the automorphisms of quantum polynomial algebras.It is generally believed that these algebras have relatively fewer automorphisms, that is, they arerigid. However, the complete understanding of their automorphisms groups is yet to be had andthe results in this paper are expected to be a step in this direction.We briefly recall the definitions. A quantum affine space O q ( F n ) over a field F is defined as O q ( F n ) := F h X , X , · · · , X n i / ( X i X j − q ij X j X i )(1)where q stands for the matrix of multiparameters q ij , that is, q = ( q ij ) and q ij ∈ F ∗ := F r { } . The matrix q is assumed to be multiplicatively anti-symmetric, that is, q ii = 1 and q ji = q − ij . Agiven quantum affine space O q ( F n ) can be embedded in a quantum torus O q (( F ∗ ) n ) by means oflocalization. The latter type of algebra is generated by the indeterminates X , · · · , X n togetherwith their inverses subject to the quantum commutation relations as in (1). The following notionplays an important role in the theory of quantum polynomials. Definition 1.1.
For the quantum polynomial algebras O q ( F n ) and O q (( F ∗ ) n ) the λ -group (denotedΛ) is the subgroup of F ∗ generated by the multiparameters q ij .For the quantum tori the monomials X m := X m · · · X m n n are units and the group-theoreticcommutator [ X m , X m ′ ] defined as[ X m , X m ′ ] := X m X m ′ ( X m ) − ( X m ′ ) − yields an alternating Z -bilinear function(2) λ : Γ × Γ → F ∗ , λ ( m , m ′ ) = [ X m , X m ′ ] , ∀ m , m ′ ∈ Γ := Z n . whose image is contained in the group Λ (e.g., [OP, Section 1]). Let(3) Λ := h p i × h p i × · · · × h p l i , l ∈ N . It was observed in [OP, Lemma 3.3(ii)] (see Section 2) that in the study of the autmorphisms ofa quantum torus the crucial case is where the group Λ is torsion-free. We may thus assume thateach direct summand h p i i ( i = 1 , · · · , l ) in (3) is an infinite cycle. For m , m ′ ∈ Γ = Z n we thus have(4) λ ( m , m ′ ) = p e ( m , m ′ )1 p e ( m , m ′ )2 · · · p e l ( m , m ′ ) l where each exponent map e i : Γ × Γ → Z yields an alternating bilinear form on Γ.It is known (e.g., [MP]) that the units of a quantum torus algebra are trivial, that is, are of theform α X m , where α ∈ F ∗ . Let us denote by A the group Aut F ( O q (( F ∗ ) n )) of all F -automorphismsand by U the group of trivial units of the algebra O q (( F ∗ ) n ). It is easily seen (e.g., [OP]) thatthe action of the group A on the quantum torus O q (( F ∗ ) n ) induces an action of this same groupon the group U of trivial units fixing F ∗ elementwise. There is thus an action of the group A onthe quotient group U / F ∗ ∼ = Γ yielding a homomorphism(5) A −→ Aut Γ = GL( n, Z )whose kernel is the group S of all scalar automorphisms defined by ψ ( X m ) = φ ( m )( X m ) for φ ∈ Hom(Γ , F ∗ ) [OP]. Thus S ∼ = ( F ∗ ) n . Furthermore by [OP, Lemma 3.3(iii)] the image of themap in (5) coincides with the group Aut( Z n , λ ) ≤ GL( n, Z ) of all non-scalar automorphisms σ ofΓ satisfying(6) λ ( σ m , σ m ′ ) = λ ( m , m ′ ) ∀ m , m ′ ∈ Z n . We thus have the following exact sequence for the group A as noted in [Ne](7) 1 → S → A → Aut( Z n , λ ) → . In terms of alternating bilinear forms e i defined above the following characterization of thenon-scalar automorphism group Aut( Z n , λ ) (as noted in [OP]) is immediate. Theorem 1.1 (Theorem 3.4 of [OP]) . Let
Sp( Z , e i ) be the symplectic group associated with theform e i . Then (8) Aut( Z n , λ ) = l \ i =1 Sp( Z , e i ) . Thus a non-scalar automorphism must preserve each of the forms e i and so must stabilize theradicals of each of these forms. This fact has been fruitfully used in [OP] towards determining thenon-scalar automorphism group Aut( Z n , λ ) in certain cases. However, notwithstanding the nicetyof the preceding theorem from a computational viewpoint (8) seems to entail some difficultieslimiting the cases where it can be directly applied.In the following we present a new approach towards the computation of this group. Beforestating it we fix some notation. Notation 1.
In view of (3) let(9) λ ( e i , e j ) = p m ( ij )1 · · · p m ( ij ) l l , ∀ ≤ i < j ≤ n, where e i , e j are standard basis vectors of the free Z -module Γ. On the (cid:0) n (cid:1) pairs ( ij ) , ( i < j ) weassume the lexicographic order. Let M ∈ Mat( n ) × l ( Z ) be the matrix whose (( ij ) , s ) entry is theexponent m ( ij ) s of p s in (9) ( ∀ s ∈ { . · · · , l } ). UANTUM POLYNOMIALS 3
With this notation we have the following.
Theorem A.
For a quantum torus O q (( F ∗ ) n ) suppose that the group Λ generated by the multi-parameters q ij is torsion-free. Set N = (cid:0) n (cid:1) and let M be as defined in Notation 1 above. Then Aut( Z n , λ ) = (cid:0) Stab
GL( n, Z ) ( M ) (cid:1) t where t denotes transposition and Stab
GL( n, Z ) ( M ) the stabilizer of M in GL( n, Z ) with respect tothe bivector representation ^ : GL( n, Z ) → GL( N, Z ) , A → ∧ ( A ) of GL( n, Z ) and the action of GL( N, Z ) on the space Mat
N,l ( Z ) by left multiplication. In otherwords Aut( Z n , λ ) = { A t | A ∈ GL( n, Z ) and ( ∧ A ) M = M } . Remark 1.1.
Although the matrix M depends on the choice of a Z -basis in the group Λ the groupStab GL( N, Z ) ( M ) (and consequently Stab GL( n, Z ) ( M )) is independent of such a choice. Indeed for achange of basis matrix P ∈ GL( l, Z ) we have M ′ = M P and then X M ′ = M ′ if and only if X M = M for X ∈ GL( N, Z ).Theorem A allows us to determine the non-scalar automorphism group Aut( Z n , λ ) thus: Step 1 : We first determine the group N := Stab GL( N, Z ) ( M ) (where N = (cid:0) n (cid:1) ). This is easilydone if M is brought to the Smith normal form. Step 2 : Next we find the subgroup N := N ∩ V (GL( n, Z )). This can be done by notingas in [Nem] that the projective transformations of the projective space P( ∧ ¯ Q n ) (here ¯ Q standsfor the algebraic closure of Q ) arising from the matrices in V (GL( n, Z )) preserve the projectivegrassmannian variety Gr(2 , n ). More algebraically, the group V (GL( n, Z )) is identified in [VP]with the connected component of the subgroup of GL( N, Z ) preserving the ideal Pl¨u generated byall the Pl¨ucker polynomials defining the variety Gr(2 , n ). Step 3 : Once N has thus been determined we must calculate N = (cid:0)V (cid:1) − ( N ). This amountsto calculating the exterior root of a matrix in N which is determined uniquely up to minussign [LoLu] and for which a polynomial-time algorithm (implemented in GAP) exists [Grn99]. Remark 1.2.
The algorithm mentioned in Step 3 can also determine (in polynomial time) if amatrix A ∈ GL( N, Z ) has an exterior root and consequently Step 2 may not be necessary. Howeverit may still be helpful in the determination of the non-scalar automorphism group.Until now the nonscalar automorphism group has been calculated ([OP, Ne]) in the case n ≥ n = 4 are given in Section 2. One such example is as follows. Example.
For n = 4 suppose that the group Λ is freely generated by the commutators q , q , q and q , while q = q = 1 . Then
Aut( Z , λ ) = ( f b,ǫ := Ü ǫ − b ǫ ǫ
00 0 0 ǫ ê , b ∈ Z , ǫ ∈ {± } ) . The following proposition is also established in Section 2.
Proposition 3.3.
Let n ≥ . For a quantum torus O q (( F ∗ ) n ) such that q = q = · · · = q n − = 1 and the remaining multiparameters independent in F ∗ we have Aut( Z n , λ ) ∼ = Z . We hope that the non-scalar automorphism group can similarly be determined in many new andinteresting situations using the approach described above. Using Theorem A we can also easilyrecover the group Aut( Z n , λ ) in the limited number of known cases [KPS94, OP, Ne] dealing withsmall values of n and the case where the group Λ has the maximal possible rank. A. GUPTA
We next consider the automorphism groups of quantum affine spaces. Until now in the study ofautomorphisms of quantum affine spaces the following important cases have been dealt with: (i)the uniparameter case, that is, q ij = q and the group h q i is infinite cyclic [AC] and (ii) the case inwhich the multiparameters are in general position [Ar], that is, the group Λ has maximum possible(torsion-free) rank. The main conclusions here are that under these assumptions the automorphismgroup shrinks to the image of the action of the torus ( F ∗ ) n . We show the same conclusion remainstrue in somewhat more general situations: Theorem B.
A quantum affine space O q ( F n ) such that the subgroup Λ has rank no smaller than (cid:0) n − (cid:1) + 1 satisfies Aut( O q ( F n )) = ( F ∗ ) n . An automorphism of the algebra O q is called linear if it stabilizes the subspace spanned by X , · · · , X n . The following proposition gives a criterion for a multiparameter quantum affine spaceto be free of linear automorphisms other than those arising from the action of the torus ( F ∗ ) n . Proposition 4.1.
Suppose that char( F ) = 2 and n ≥ . Let q = ( q ij ) be a multiplicativelyantisymmetric matrix whose entries satisfy(i) q ij = q kl ∀ i < j, ∀ k < l, (ii) q ij q kl = 1 ∀ i < j, ∀ k < l, ( i, j ) = ( k, l ) . The group of linear automorphisms of the quantum affine space O q ( F n ) coincides with the torus ( F ∗ ) n . Notation 2.
We will use the short forms O q for the quantum affine space and “ O q for quantumtorus obtained from it by localizing at the multiplicative subset generated by the indeterminates X i (1 ≤ i ≤ n ) and refer to it as the “corresponding quantum torus”. Automorphisms of thealgebras we consider will always be F -automorphisms.2. Automorphisms of quantum tori
As noted above in (7) the essential question here is the determination of the group Aut( Z n , λ ) ≤ GL( n, Z ) of non-scalar automorphisms. The following fact shown in [OP] is the basis for ourassumption that the group Λ is torsion-free. Lemma 2.1 (Lemma 3.3(ii)) . Let p denote the size of the torsion subgroup of Λ . The subalgebra c O ′ of “ O q generated by the powers X ± pi of the indeterminates X i is a characteristic sub-algebra ofthe same rank. Moreover “ O q is free left c O ′ -module of finite rank and the corresponding λ -group Λ ′ associated with c O ′ is torsion free. In view of Lemma 2.1 we will assume throughout this section that the group Λ is torsion-free.We recall that for a given matrix A ∈ GL( n, Z ) the exterior square ∧ A of A is the (cid:0) n (cid:1) × (cid:0) n (cid:1) -matrixwhose rows and columns are indexed by the pairs ( ij ) (1 ≤ i < j ≤ n ) ordered lexicographicallyand whose (( ij ) , ( kl )) entry is the 2 × i, j and columns k, l .By the well-known Cauchy-Binet formula the map A
7→ ∧ A is multiplicative and satisfiesdet( ∧ A ) = (det A ) n − [HJ]. We also have ∧ A t = ( ∧ A ) t whete t denotes transposition. With M as defined in Notation 1 we have the following. Theorem A.
For a quantum torus O q (( F ∗ ) n ) suppose that the group Λ generated by the multi-parameters q ij is torsion-free. Set N = (cid:0) n (cid:1) and let M be as defined in Notation 1 above. Then Aut( Z n , λ ) = (cid:0) Stab
GL( n, Z ) ( M ) (cid:1) t where t denotes transposition and Stab
GL( n, Z ) ( M ) the stabilizer of M in GL( n, Z ) with respect tothe bivector representation ^ : GL( n, Z ) → GL( N, Z ) , A → ∧ ( A ) of GL( n, Z ) , that is, Stab
GL( n, Z ) ( M ) = { A ∈ GL( n, Z ) | ( ∧ A ) M = M } . UANTUM POLYNOMIALS 5
Proof.
Writing the group Λ ≤ F ∗ additively, in view of Notation 1 we have(10) λ ( e i , e j ) = m ij ) p + · · · + m l ( ij ) p l , ∀ ≤ i < j ≤ n, where m ( ij ) k ∈ Z . Thus M = ( a ( ij ) s ) where a ( ij ) s = m ( ij ) s . Now let M = ( m ij ) ∈ GL( n, Z )be such that M t ∈ Aut( Z n , λ ). Setting e ′ j = M t e j = m j e + · · · + m jn e n we note that since λ is an alternating function therefore λ ( e ′ i , e ′ j ) may be expressed (e.g., [MP,Section 1.2]) in terms of λ ( e i , e j ) where i < j as follows:(11) λ ( e ′ i , e ′ j ) = m ij, λ ( e , e ) + m ij, λ ( e , e ) + · · · + m ij, ( n − n λ ( e n − , e n ) . We also note that the coefficients appearing in the RHS of the above expression constitute row (ij)of the matrix ∧ ( M ). Since M t is λ -preserving by (6) we have λ ( e ′ i , e ′ j ) = λ ( e i , e j ) ∀ ≤ i < j ≤ n. Expanding and comparing the coefficients of p s ( s = 1 , · · · , l ) in both sides of the last equation weget noting(12) m ij, m (12) s + m ij, m (13) s + · · · + m ij, ( n − n m (( n − n ) s = m ( ij ) s ∀ s = 1 , · · · , l. Letting ( ij ) take values in the set(13) { (12) , (13) , · · · , ( n − n } in equation (12) and setting M ( s ) = ( m (12) s , m (13) s , · · · , m (( n − n ) s ) t we thus get ∧ ( M ) M ( s ) = M ( s ) ∀ s ∈ , · · · , l. As M ( s ) is column s of M (as defined in Notation 1) it follows that: ∧ ( M ) M = M . Clearly the above reasoning is reversible. This establishes the assertion of the theorem. (cid:3)
Remark 2.1.
In the situation of Theorem A since the N elements λ ( e i , e j ) (1 ≤ i < j ≤ n )generate the group Λ therefore the columns of M must span a free Z -submodule of Z N of rank l .Thus M must fix a free submodule of Z N of rank l .The remainder of this section is devoted to the determination of the automorphism group ofthe quantum torus defined by multiparameters satisfying a given set of conditions. Our line ofapproach as suggested by Theorem A for determining the matrices A ∈ GL( n, Z ) whose transposelies in the group Aut( Z n , λ ) can be summarized as follows.(14) Relations matrix M −→ Submodule Fixed by ∧ A −→ ∧ A −→ A .
Example 1.
For n = 2 , the group Λ is infinite cyclic with the generator q = q i and M is the × matrix . Clearly, the stabilizer of M is { } and thus Aut( Z , λ ) = SL(2 , Z ) . Example 2 ([OP]) . For n = 3 the group Aut( Z , λ ) was discussed in [OP] . The most non-trivialsituation is obtained when Λ ∼ = Z . As shown in the proof of [OP, Proposition 3.7] using a changeof variables we may suppose in this case that q = 1 and the set { q , q } is independent where q ij = [ X i , X j ] and moreover Aut( Z , λ ) = ( Ñ ǫ a ǫ b ǫ é , a, b ∈ Z , ǫ ∈ {± } ) . Note that in this case M := Ñ é and therefore for A ∈ Aut( Z , λ ) , ∧ A t = Ñ ∗ ∗ é . A. GUPTA
We also note that ∧ Ñ ǫ a ǫ b ǫ é t = Ñ bǫ − aǫ é as would be expected from Theorem A. As just seen in Example 2 for n = 3 and Λ ∼ = Z a change of variables leads to a simple set ofrelations for the commutators [ X i , X j ]. In turn this means a simple form for the relations matrix M in Theorem A thus facilitating the computation of the non-scalar automorphism group in thiscase.The situation is more complex for n = 4. For example, if n = 4 and Λ ∼ = Z one msy expectthat a suitable change of variables will allow us to assume, for example, that q = 1. Howeverthis is not true by the example of [MP, Section 3.11] and is still not true when Λ ∼ = Z [GQ]. Proposition 2.2.
For n = 4 suppose that the group Λ is freely generated by the commutators [ X , X ] , [ X , X ] , [ X , X ] and [ X , X ] , while [ X , X ] = [ X , X ] = 1 . Then
Aut( Z , λ ) = ( f b,ǫ := Ü ǫ − b ǫ ǫ
00 0 0 ǫ ê , b ∈ Z , ǫ ∈ {± } ) . Proof.
In this case for A t ∈ Aut( Z , λ ) using Theorem A we see that the columns C ( i ) , i = 1 , · · · , ∧ A satisfy C ( i ) = I ( i ) , ∀ i = 3 , · · · , I ( i ) stands for the i -th column of the identitymatrix I . Thus only the first two columns need to be determined. To this end we note that by[Nem] the image of ∧ A under the projection ρ : GL(6 , ¯ Q ) → PGL(6 , ¯ Q ) preserves the projectivegrassmannian variety Gr(2 ,
4) embedded in the projective space P( ∧ ¯ Q ) where ¯ Q stands for thealgebraic closure of Q .By a well-known fact [Be, Section 14.7] we know that the group of the projective quadricGr(2 ,
4) is the image of the isometry group O( β ) in the projective group PGL(6 , ¯ Q ) where β is thepolarization of the Pl¨ucker quadratic form q ( ξ , ξ , ξ , ξ , ξ , ξ ) = ξ ξ − ξ ξ + ξ ξ . It is easily checked that the matrix of the function β with respect to the basis e i ∧ e j ( i < j ) isthe matrix P := − − . Thus ( ∧ A ) C ∈ O( β ) for some scalar matrix C ∈ GL(6 , ¯ Q ). Therefore ∧ A must satisfy(15) C ( ∧ A ) t P ∧ A = P. As det ∧ A = (det A ) = ± C = ±
1. Writing C = diag( λ, · · · , λ ) this meansthat either λ is root of the polynomial Φ − := Y − + := Y + 1 = 0. Since the matrices ∧ A and P have integer entries it is clear from (15) that λ = ± C = ± I . This means that ∧ A ∈ O( β ). Direct calculation using (15) reveals that ∧ A has the form ∧ A = b b , b ∈ Z . UANTUM POLYNOMIALS 7
As is readily checked the last equation means noting [LoLu, Corollary 2] that A = ǫ Ü − b ê , ǫ = ± . and thus Aut( Z , λ ) = ( f b,ǫ := Ü ǫ b ǫ ǫ
00 0 0 ǫ ê , b ∈ Z , ǫ ∈ {± } ) . (cid:3) Example 3.
Following a similar approach as in Proposition 2.2 we can show that for n = 4 assuming that the group Λ is freely generated by the commutators [ X , X ] , [ X , X ] and [ X , X ] ,while [ X , X ] = [ X , X ] = [ X , X ] = 1 the non-scalar automorphism group is given in this case by Aut( Z n , λ ) = ( φ a,b,ǫ := Ü ǫ a b ǫ ǫ
00 0 0 ǫ ê , a, b ∈ Z , ǫ ∈ {± } ) . Remark 2.2.
In a more general situation where M does not have a simple form as seen in theabove examples it is easily checked thatStab GL( N, Z ) ( M ) = U − (Stab GL( N, Z ) ( U M V ) U where U M V is the Smith normal form SN F ( M ) of M . Proposition 2.3.
For n = 4 suppose that the multiparamters q ij ( i, j ) = (1 , are independentfor i < j and q = Y i Aut( Z , λ ) = {± I } .Proof. Clearly, in this case the relations matrix M has the form M = Using the smith form calculator [Ma] we find SN F ( M ) = U M V = , U = − , V = I . Clearly, Stab GL( N, Z ) ( U M V ) = x y z u v , x, y, z, u, v ∈ Z . A. GUPTA Using Remark 2.2 and calculating with the help of SageMath [SM] we find thatStab GL( N, Z ) ( M ) = − S S S S S S − x x + 1 x x x x − y y y + 1 y y y − z z z z + 1 z z − u u u u u + 1 u − v v v v v v + 1 , S = u + v + x + y + z. For a matrix B ∈ V (GL( n, Z ) ∩ Stab GL( N, Z ) ( M ) by the same reasoning as in Proposition 2.2we obtain B t P B = P . Comparing the first rows in both sides we obtain T − v = 0 u + v − T = 0 v − z − T = 0 v − y − T = 0 v + x − T = 0 u + x + y + z + T = 0where T = 2 uv + 2 v − ux + 2 vx + 2 vy + 2 vz + 2 yz . It is easily seen that this system has a uniquesolution x = y = z = u = v = 0. This completes our proof. (cid:3) It was shown in [OP] that if the (cid:0) n (cid:1) multiparameters q ij (1 ≤ i < j ≤ n ) are independent in F ∗ then Aut( Z n , λ ) ∼ = Z . With the help of Theorem A we show in the next proposition that thesame conclusion remains valid under a somewhat weaker hypothesis. Proposition 2.4. For a quantum torus O q (( F ∗ ) n ) with the n − multiparameters q , q , · · · , q n − set to one and the remaining multiparameters q n − , q n , q , q , · · · , q ( n − n independent in F ∗ Aut( Z n , λ ) ∼ = Z and consequently by (7) we have → Hom( Z n , F ∗ ) → Aut( O q ( F ∗ n )) → Z → . Proof. From the theorem hypothesis it is evident that in this case the relations matrix M of Theo-rem A is obtained from the identity matrix I ( n ) by deleting the first n − B ∈ Stab( M ) must coincide with I ( n ) in all but the first n − B musthave the form(16) B = á ∗ · · · ∗∗ · · · ∗ ∗ · · · ∗ I ( n − ) ë . Suppose that B = ( b ij,kl ) is induced from a matrix A := ( a uv ) ∈ GL( n, Z ), that is, B = ∧ ( A ).The ( n − × ( n − A + of A formed by the rows 2 , · · · , n and columns 2 , · · · , n satisfies ∧ ( A + ) = I ( n − ). As is well knowndet( ∧ ( A + )) = det( A + ) n − and hence A + is also nonsingular. Applying [LoLu, Corollary 2] we obtain(17) A + = ǫI n − , ǫ ∈ {− , +1 } . In view of (16) we clearly have b , j = 0 ∀ j = 3 , · · · , n. UANTUM POLYNOMIALS 9 Computing the 2 × A corresponding to the (12 , j ) entries of B with the help of (17) itis immediately seen that the 1 j entry a j of A equals to zero. Similarly, b , = 0 implies a = 0.By the same token using the evident fact that b jn, n = 0 ∀ j = 2 , · · · , n − a j = 0. Again b ( n − n, n − = 0 implies a n = 0.It now follows that A is diagonal and moreover A ∈ GL( n, Z ) implies(18) a = ǫ ′ , ǫ ′ ∈ {− , +1 } But ǫ = ǫ ′ as otherwise b (1 n ) , (1 n ) = ǫ ′ ǫ = − B as noted in (16). Thus A = ǫI n and the proof is complete. (cid:3) Automorphisms of quantum affine spaces - preliminary facts Automorphisms of the quantum affine spaces O q were considered in [AC, Ar, OP]. In ourtheorems to follow we shall be utilizing the definitions, facts and results in these articles which webriefly recall here and also add a few easy propositions. Definition 3.1 (Section 1.4 of [AC]) . An automorphism σ of O q is called linear if it has the form(19) σ ( X i ) = n X j =1 α ij X j ∀ i ∈ { , · · · , n } , ( α ij ) ∈ GL( n, F ) . In other words, an automorphism is linear if and only if it preserves the degree one componentof the algebra O q with respect to the N -grading by total degree. The subgroup of linear auto-morphisms is denoted as Aut L ( O q ). For a matrix ( α ij ) ∈ GL( n, F ) to define an automorphism asin (19) the following necessary and sufficient conditions must hold ([AC]):(20) α ik α jl (1 − q ij q lk ) = α il α jk ( q ij − q lk ) ∀ i < j, ∀ k ≤ l. The last equation may be re-written as(21) α ik α jl ( q kl − q ij ) = α il α jk ( q kl q ij − ∀ i < j, ∀ k ≤ l. Setting k = l in the last equation we obtain(22) α ik α jk (1 − q ij ) = α ik α jk ( q ij − ∀ i < j, ∀ k ∈ { , · · · , n } . Observation. Clearly, the last equation means that if char( F ) = 2 and none of the multiparam-eters q ij ( i < j ) equals to unity then at least one of the coefficients α ik and α jk vanishes. It isimmediate that in this case the nonsingular matrix ( α ij ) has exactly one nonzero entry in eachrow and each column.The next proposition is an easy consequence of the preceding observation. Proposition 3.1. Suppose that char( F ) = 2 and the entries of q satisfy ( (cid:7) ) q ij = 1 ∀ ≤ i < j ≤ n. Then (23) Aut L ( O q ) ∼ = F ∗ n ⋉ P for a subgroup P of S n . Besides the rather simple condition in ( (cid:7) ) there are other situations where (23) holds. Indeedas shown in [OP] this is the case if the following condition is satisfied:( ♦ ) The localization “ O q of O q has center F . In view of the foregoing we make the following definition. Definition 3.2. Given a quantum affine space O q a permutation π ∈ S n is said to be admissible if π ∈ Aut( O q ). By the remark following [OP, Proposition 3.2] a permutation π is admissible if and only if theany of the following equivalent pair of conditions holds for the corresponding permutation matrix p .(24) pqp t = q ⇔ pq = qp . The next lemma and proposition record some consequences of a permutation π being admissible. Lemma 3.2. For a given quantum affine space O q if a permutation π is admissible then (i) for each r -cycle ( j j · · · j r ) , where r ≥ in the decomposition of πq j r j = q j j = q j j = · · · = q j r − j r − = q j r − j r , and (ii) for each fixed point k of π and each r -cycle ( j j · · · j r ) in the the decomposition of πq j k = q j k = · · · = q j r k . Proof. (i) This follows easily by applying π to X j t X j t ˙+1 = q j t j t ˙+1 X j t ˙+1 X j t where ˙+ denotes addition modulo r . As a result we obtain X j t ˙+1 X j t ˙+2 = q j t j t ˙+1 X j t ˙+2 X j t ˙+1 for all t ≥ r . Part (ii) is similar to (i). (cid:3) Proposition 3.3. Suppose that the group Aut( O q ) contains a non-trivial permutation π . Thenthere exists a relation (25) q ij q kl = 1 , ( i, j ) , ( k, l ) ∈ { , · · · , n } such that i < j and k < l .Proof. If ( j j · · · j r ) is a cycle in the decomposition of π then by Lemma 3.2 we have(26) q j r j = q j j = q j j = · · · = q j r − j r − = q j r − j r , Evidently the sequence of differences j t ˙+1 − j t in (26) where t varies modulo r in the set { , · · · , r } has both positive as well as negative terms. We can thus find u, v ( u = v ) and such that j u ˙+1 − j u and j v ˙+1 − j v have opposite signs. Without loss of generality we may assume that j u +1 − j u < q j v j v +1 q j u +1 j u = 1 . (cid:3) Combining the above facts immediately yields a criterion for ensuring that all linear automor-phisms of a quantum affine space result from the action of the torus. Corollary 3.4. Suppose that char( F ) = 2 and the multiparameters satisfy: ( ♣ ) q ij q kl = 1 for all i < j and k < l. Then Aut L ( O q ) = ( F ∗ ) n . Proof. The condition ( ♣ ) means that q ij = ± i < j . Thus Proposition 3.1 applies andwe conclude by Proposition 3.3. (cid:3) In the next section we will show that the same conclusion remains valid under a differenthypothesis which allows 1 to be included as an (off-diagonal) entry of q . UANTUM POLYNOMIALS 11 Automorphisms of quantum affine spaces–Main results Proposition 4.1. Suppose that char( F ) = 2 and n ≥ . Let q = ( q ij ) be a multiplicativelyantisymmetric matrix whose entries satisfy(i) q ij = q kl ∀ i < j, ∀ k < l, (ii) q ij q kl = 1 ∀ i < j, ∀ k < l, ( i, j ) = ( k, l ) . The group of linear automorphisms of the quantum affine space O q ( F n ) coincides with the torus ( F ∗ ) n .Proof. Note that the hypothesis means that there is at most one entry q i ′ j ′ with i ′ < j ′ that isequal to 1. Similarly, there is at most one entry q i ′′ j ′′ with i ′′ < j ′′ that is equal to − 1. Supposethat A = ( α ij ) ∈ GL( n, F )induces a linear automorphism of the given quantum affine space. It suffices to show that A is adiagonal matrix. Using (22) it is easily seen that in any column of A at most two entries can benonzero and in the case there are two nonzero entries, these must be in the i ′ -th and j ′ -th rows.We claim that there can be at most two columns in A that have two nonzero entries. Indeedsuppose that 2 + s columns have two nonzero entries necessarily in the i ′ -th and j ′ -th rows. Asjust noted the remaining n − − s columns each has exactly one non-zero entry. Clearly, if s > n − − s nonzero entries in A cannot fulfill the requirement of a non-zero entry in each ofthe n − i ′ -th and j ′ -th rows. This shows that s = 0.Next we note that at least one of the entries α i ′ i ′ and α j ′ i ′ is non-zero. To see this we pick m < p in the range 1 , · · · , n . By (21) we have noting that q i ′ j ′ = 1 α i ′ m α j ′ p ( q mp − 1) = α i ′ p α j ′ m ( q mp − . (27)If ( m, p ) = ( i ′ , j ′ ) then as noted above q mp = 1 and (27) reveals that the minor of A correspondingto the 2 × i ′ -th and j ′ -th rows and the m -th and p -th columns is equalto zero. If α i ′ i ′ = α j ′ i ′ = 0 then the minor corresponding to the 2 × K defined by the i ′ -th and j ′ -th rows and the i ′ -th and j ′ -th columns is also equal to zero. This would mean that arow of the exterior square ∧ A of A is the zero row contradicting the assumption A is non-singular.By the same token at least one of the entries α i ′ j ′ and α j ′ j ′ in column j is nonzero. Moreover, thenon-zero entries in the two columns, namely, i ′ and j ′ cannot be in only one of the rows i ′ or j ′ asin this case the determinant of K will be zero.We now claim that if a column of A has two non-zero entries then it must be the i ′ -th or the j ′ -th column. Indeed let h be a column of A having two non-zero entries. As noted above thesenon-zero entries of h must be rows i ′ and j ′ . Thus the three columns i ′ , j ′ and h have non-zeroentries only in rows i ′ and j ′ . Consequently there can be at most n − i ′ , j ′ and h that are contained in the n − i ′ and j ′ . But this meansthere is a row with no non-zero entry contradicting the assumption that A is non-singular.We now consider a column of A , say the k -th, that has only one non-zero entry, say in the u -throw. This is possible in view of the foregoing noting that n ≥ α uk be the unique nonzero entry in the k -th column. In view of the foregoing observationsit is easily seen that α uk is also the unique nonzero entry in the u -th row. We note as n ≥ l (1 ≤ l = k ≤ n ) such that either( i ) k < l and ( k, l ) = ( i ′′ , j ′′ ); or , ( ii ) l < k and ( l, k ) = ( i ′′ , j ′′ ) . (28)note that we do not assume that the column l has only one non-zero entry. Step 1: In case (i) we suppose that the l -th column has a nonzero entry, namely, α vl in the v -throw. We claim that v > u . Indeed if v < u then (21) yields(29) α vk α ul ( q kl − q vu ) = α uk α vl ( q kl q vu − . By the preceding paragraph the LHS of (29) vanishes (as α uk is the only non-zero entry in column k ) and (noting theorem hypothesis) the RHS is nonzero, unless,(30) ( k, l ) = ( v, u ) = ( i ′′ , j ′′ ) . But by the choice of the pair ( k, l ) (30) cannot hold. Our assertion, namely, u < v now follows.We now apply (21) to u < v and k < l and thus obtain α uk α vl ( q kl − q uv ) = α vk α ul ( q kl q uv − . Clearly in the last equation the RHS is equal to zero, but the LHS can vanish only if q kl = q uv andby the theorem hypothesis this is possible only if k = u and l = v . The k -th and l -th columns of A must therefore coincide with the corresponding columns of a suitable diagonal matrix D . Step 2: We now suppose that case (ii) holds in (28). As before let α vl = 0 for some v . Weclaim that in this case v < u . For assuming the contrary and applying (21) leads to a contradictionquite similarly to that resulting from (29) above. Arguing as in case (i) and applying (21) to v < u and l < k we obtain α vl α uk ( q lk − q vu ) = α ul α vk ( q lk q vu − l = v and k = u . Conclusion of proof: Suppose that k = i ′′ , j ′′ . Then Steps 1 and 2 above immediately yieldthe fact that the matrix A is a diagonal. Otherwise if k = i ′′ (resp. k = j ′′ ) by the above reasoningwe obtain that the s -th column of A has the form α s e s (where e s stands for the s -th column ofthe identity matrix and α s ∈ F ∗ ) except possibly when s = j ′′ (resp. s = i ′′ ). But we may nowredefine k to be any natural number in { , · · · , n } \ { i ′′ , j ′′ } and apply one of the Steps 1 and 2depending on whether k < j ′′ or k > j ′′ (resp. k < i ′′ or k > i ′′ ). (cid:3) As before, let { e , · · · , e n } denote the standard basis in Γ := Z n . Then { e i ∧ e j | ≤ i < j ≤ n } is a basis in ∧ Γ. As usual, for a permutation π ∈ S n let P ∈ GL( n, Z ) denote the correspondingpermutation matrix. Clearly S n acts on the Z -module ∧ Γ via(31) π ( e i ∧ e j ) = ∧ P ( e i ∧ e j ) = e π ( i ) ∧ e π ( j ) ∀ π ∈ S n . In this action the image of π in End Z ( ∧ Γ) will be denoted as ∧ π . By restriction we get an actionof S n on the subset ¯ B = { ǫe i ∧ e j | i < j, and ǫ ∈ {− , }} . We note the following lemma that will be used in the proof of Theorem B. Lemma 4.2. Let π ∈ S n ( n ≥ and Fix( ∧ π ) stand for the sub-module of ∧ Γ left fixed by thepermutation π under the action (31) . Then rk(Fix( ∧ π )) ≤ Ç n − å . with equality holding whenever π a transposition.Proof. It is seen without difficulty that the submodule Fix( ∧ π ) is generated by the sums ofelements in the orbits of the cyclic subgroup C π generated by π acting on the set ¯ B as definedin the discussion preceding the theorem. Needless to say such “orbit sums” need not be linearlyindependent (over Z ) or even be non-zero. For example the C ( ij ) -orbit { e i ∧ e j , − e i ∧ e j } has zerosum. Let N π the number of C ( ij ) -orbits in this action on the set ¯ B . By an application of thewell-known formula of Burnside it can be shown ([Si]) that the number N π is maximal when π isa transposition ( ij ) and in this case N ( ij ) = ( n − n − 3) + (2 n − . It is also clear that the orbit C ( ij ) ( e i ′ ∧ e j ′ ) has a nonzero sum whenever ( i, j ) = ( i ′ , j ′ ) and thatthis sum is the negative of the sum of the orbit C ( ij ) ( − e i ′ ∧ e j ′ ). We thus obtainrk (cid:0) Fix( ∧ ( ij )) (cid:1) = N ( ij ) − 12 = Ç n − å Z -independent orbit-sums that constitute a Z -basis for Fix( ∧ π ). In general the group C π willhave some orbits whose sum is zero. Clearly these orbits are precisely the orbits C π ( e i ∧ e j ) suchthat C π ( ± e i ∧ e j ) ∋ ∓ e i ∧ e j and the orbits C π with non-zero sum are precisely those for which C π ( e i ∧ e j ) ∩ C π ( − e i ∧ e j ) = ∅ . It is immediate from this that the number of independent orbit-sums for the permutation π arebounded above by (cid:4) N π (cid:5) .Evidently for any non-identity permutation π ∈ S n the number of fixed points Fix( π ) is boundedabove by n − ij ). Using this fact and the UANTUM POLYNOMIALS 13 Burnside formula an upper bound for N π was obtained in [Si] as follows N π = 1 | C π | n ( n − 1) + X φ ∈ C π ,φ =1 Ç Fix( φ )2 å ! ≤ n ( n − | C π | + ( | C π | − | C π | Ç n − å = ( n − n − 3) + 4 n − | C π | . This yields N ( ij ) − N ( π ) ≥ (4 n − − | C π | ! and thus for n ≥ (cid:0) ∧ ( ij )) (cid:1) − rk(Fix (cid:0) ∧ π ) (cid:1) = N ( ij ) − − õ N π û ≥ − 12 + N ( ij ) − N π ≥ − 12 + (2 n − − | C π | ! ≥ − . As N ( ij ) − , (cid:4) N π (cid:5) ∈ Z it clearly follows thatrk(Fix (cid:0) ∧ ( ij )) (cid:1) ≥ rk(Fix (cid:0) ∧ π ) (cid:1) . (cid:3) Theorem B. A quantum affine space O q ( F n ) such that the subgroup Λ has rank no smaller than (cid:0) n − (cid:1) + 1 satisfies Aut( O q ( F n )) = ( F ∗ ) n . Proof. As usual we write O q = O q ( F n ). The assumption concerning the rank of the group Λ meansthat the corresponding quantum torus “ O q (arising from localization) of the given quantum affinespace O q ( F n ) has center Z no bigger than the ground field F . Indeed viewing the latter algebra asa crossed product F ∗ Γ (Γ = Z n ) as in [OP] it is easily seen that for any subgroup Γ ′ of Γ of finiteindex the corresponding λ -group Λ ′ of the sub-quantum torus F ∗ Γ ′ has the same rank as that ofthe group Λ.It is well known (e.g., [OP]) that the center of a quantum torus is a generated by monomials.Given a central monomial z := X m in the algebra “ O q we can clearly extend the set { z } to a set of n monomials which together with their inverses generate a subalgebra of the form F ∗ Γ ′ as describedin the previous paragraph. But the central monomial z evidently reduces rk(Λ ′ ) by n − ≤ (cid:0) n − (cid:1) and thus contradicting the theorem hypothesis. We also note that by the theoremof Section 1.3 of [MP] the algebra O q is simple.Let π be a permutation of the generators X , · · · X n of the generators of the algebra “ O q . To showthe assertion in the theorem it suffices in view of [OP, Propositon 3.2] to show that if π ∈ Aut( O q )then π = id. To this end we note that by [OP, Proposition 1.5] the permutation π extends to anautomorphism of the quantum torus “ O q . Moreover this extension induces on Γ the automorphismwhich is given by the permutation matrix P corresponding to π . By Remark 2.1Fix ∧ π ≥ Ç n − å + 1contradicting Lemma 4.2 (cid:3) Examples.Example 4. In the situation of Example 2 of Section 2 we consider the quantum affine spacedefined by the same matrix q of multiparameters. In this case the λ -group has rank and soTheorem B applies. In particular this quantum space is a simple ring and whose automorphisms(permutations of generators) lift to automorphisms of the corresponding quantum torus “ O q . Asseen in the same example referred to above these automorphisms induce on Γ automorphisms havingthe form Ñ ǫ a ǫ b ǫ é But the only permutation matrix which has this form is the identity. In otherwords the assertion of Theorem B holds true in this case. Example 5. Consider the multiplicatively antisymmetric matrices q ∈ M ( F ) defined by the × grid of Figure 1 where the boxes with the same pattern indicate that the corresponding entries of Figure 1. Multiplicatively antisymmetric matrix q with circulant symmetry q are equal. Furthermore, the (equal) entries in the boxes with northeast hatching are inverses ofthe (equal) entries in the boxes with northwest hatching. Similarly for the boxes with vertical andhorizontal hatching. The unhatched boxes correspond to ∈ F .Thus the matrices arising in this way are multiplicatively antisymmetric as well as circulant.Evidently, the rank of the λ -group in this case is at most .Using (24) it is easily seen that the the 5-cycle (12345) is an automorphism of O q which clearlylifts to an automorphism of the corresponding quantum torus “ O q (and thus induces an automor-phism of Z ). As rk(Fix( ∧ (12345)) = 2 this is agreement with Remark 2.1. UANTUM POLYNOMIALS 15 References [AC] J. Alev and M. Chamarie, Derivations et automorphismes de quelques algebras quantiques, Communicationsin Algebra, 20:6, 1787-1802,[Ar] V. A. Artamonov, Quantum polynomial algebras, J. Math. Sci. (New York) 87 (1997), no. 3, 3441–3462.[AW] V. A. Artamonov and R. Wisbauer, Quantum polynomial algebras, Algebr. Represent. Theory 4 (2001), no.3, 219–247.[Be] M. Berger, Geometry II, Springer.[Br] C. J. B. 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