CC BY
22
UDC 512.743
A NECESSARY CONDITION FOR THE ELEMENTARY MATRIX GROUP TO BE LINEAR OVER A FIELD
G.A. Noskov
d.f.-m.n., s.r., e-mail: g.noskov@googlemail.com
Institute of Mathematics, SORAN
Abstract. We prove that if R is an associative unital ring and the elementary group En(R) for n > 3 is linear over a field k of characteristic zero, then R has a finite index ideal which is linear over k. We prove that if A is an infinite integral domain of characteristic p > 0, then for every natural n the ring of Witt vectors Wn(A) is not virtually linear over any field. However, somewhat paradoxically, for any field k and any m,n > 1 the group GLm (Wn (k)) is linear over k.
Keywords: linear group, field, affine algebraic group, associative ring, Witt ring.
Introduction
This paper is a complement to the studies reported earlier by M. Kassabov and M. Sapir [8]. Recall that a general linear group GLn(R) over (always an associative and unital) ring R consists of all invertible n x n matrices over R with operations of matrix multiplication and inversion. A matrix (or linear) group over R is an abstract group embeddable into GLn(R) for some n > 1. Similarly, a ring A is linear over R if it can be embedded into the ring of n x n matrices Mn (R) over the ring R for some n > 1. A fundamental question in group theory is to determine whether a given group G is linear over some (or certain) field or not. Here we study the case of the elementary group G = En(R) over a ring R. Recall that En(R) is the subgroup of GLn(R) generated by all elementary (n x n)-matrices Xij (r) = Id + reij (r e R, 1 < i = j < n) , where Cij is a standard matrix unit with 1 in the (i,j)-position, and 0's elsewhere. The group En(R) is certainly linear over R, so the crux of the matter is whether or not En(R) is linear over some field. Our main result is
Theorem 1. Let R be an associative unital ring. If group En(R) for n > 3 is linear over a field k of characteristic zero, then R is almost linear over k, i.e. R has a finite index ideal I, which is linear over k .
The theorem should have been known for many years, but it has been proved only very recently in case k = C [8]. Informally, our theorem can be considered as a manifestation of the Lefschetz principle applied to the theorem by Kassabov and
Sapir. Recall that the "Lefschetz principle" states that any sentence in the firstorder language of fields which is true for complex numbers is also true for every algebraically closed field of characteristic 0. The linearity of En(R) is definitely not a first-order sentence in the language of fields (although we were not able to prove this).
The study of isomorphic representability of infinite groups by matrices was initiated by Mal'cev (1940) in a paper in which he found the conditions for the representability of abelian and periodic groups (over a field) and proved a local theorem for matrix representability of bounded degree [10]. The state of art during the period 1966-1977 is surveyed in [13,14]. A lot is known about isomorphisms between various matrix groups over (mostly commutative ) rings; see [17,22]. See also [3,4,6]. Moody, Long and Paton, Krammer, Bigelow made a remarkable progress in the linearity problem for braid groups [9,16,19,20] . For the recent studies see [21,23].
In the proof of Theorem 1 in [8, Thm.1] (in case k = C) a nice topological argument plays a central role. Namely, the set Z = 1 + Re1n is an abelian subgroup of En(R) and at the same time Z has a natural ring structure isomorphic to R. Suppose En(R) is linear over C . Identifying En(R) with its image in some GLn(C), consider Z — the closure of Z in the Zariski topology on GLN(C). The first observation is that Z possesses the structure of an associative algebraic ring with unit, which extends the R-ring structure on Z . The addition operation is given by the matrix multiplication on GLN(C). The Zariski connected component
of the additive group Z° is a two-sided ideal in Z. The key fact is that
is isomorphic to C^ for some natural m. Here the authors of [8] apply topological
argument, calculating the fundamental group of ^Z°, +.
In this paper, we avoid the reference to topology. With some small amount of the theory of algebraic groups, we show that Z is a unipotent group (see Section 1.). Moreover, to prove unipotence, we even do not use linearity of En(R) - it is enough to make use of linearity of the 2-step nilpotent Heisenberg group UT3(R).
Clearly, if R is linear over a field, then the group En(R) also is linear over the same field for any n > 2. It is stated in [8, (Thm.1)] that the conclusion is true even if R is virtually linear over a field. Kassabov and Sapir also manage to build an example of a "strange" commutative ring R which is not linear over any field (even virtually) but for which En(R) is linear over field. We study their example from the scientific point of view and we show that any truncated Witt ring Wn (k) satisfies the above property for every natural n.
1. The Heisenberg group over a ring
For any associative unital ring R and any natural n > 2 let UTn(R) denote the upper uni-triangular group over R of the size n x n. By this we mean the group
of n n matrices
UTn(R)
In case n = 3 we have
(l R ••• R\
0 .
0 0 ... R
0 0 0 1
UTa(R)
1R
R
01R 0 0 1
and the matrix multiplication looks like this:
X z\ /1 x' z'\ (1 X + x' xy' + z + z'^
0 1 y' 0 0 1
0 1 y 0 0 1
0
/
1 0
y + y' 1
where the entries x,...,z are arbitrary elements of R. Identifying the first two matrices with the raws (x, y,z), (x',y',z'), we can rewrite the multiplication law as follows:
(x, y, z) (x', y', z') = (x + x', y + y', xy' + z + z').
The associativity can be easily verified, the triple (0, 0, 0) is an identical element and the inverse is given by
(x,y,z)-1 = (-X, -y,xy - z).
We call U = UT3(R) the Heisenberg group over R.
Lemma 1. The subgroup Z = (0,0,R) is the center of U and at the same time the commutator subgroup of U. Moreover, the commutator map U x U ^ Z is surjective.
Proof. As usual, we denote by [g,h] the group commutator g-1h-1 gh of the group elements g, h. It follows from the formula
(x, y, z) (0, 0, z') = (0, 0, z') (x, y, z) = (0, 0, z + z')
that Z lies in the center. On the other hand, if (x, y, z) is in the center, then
[(x, y, z) , (1, n, 0)] = (0, 0, x - ny) = 0 (1)
for all natural n. It follows from x — y = 0,x — 2y = 0 that x = 0,y = 0 and thus (x,y,z) e Z. The commutator formula
[(x,y,z), (x',y',z')] = (0, 0, xy' — x'y) (2)
shows that the commutator subgroup [U, U] lies in Z. Setting x = 1,x' = 0 in the above formula, we obtain
[(1,y,z)(0,y',z')] = (0, 0, y'), (3)
which shows that Z C [U, U] and finally Z = [U, U].
■
Hence U is a 2-step nilpotent group. The sets X = (R, 0, 0) , Y = (0, R, 0) are abelian subgroups and XZ, XZ are normal abelian subgroups in U. We consider Z as a ring isomorphic to R.
2. Abstract linear representations of the Heisenberg group
The main result of this section is
Theorem 2. Let R be a ring (associative, unital), k a perfect field, m > > 3,n > 2 - the integers. Let p : UTm(R) ^ GLn(k) be an (abstract) injective homomorphism. Then for any i, j with j — i > 2, the closure of p (xj (R)) in the Zariski topology on GLn(k) is a virtually unipotent k-group.
We need some preliminaries for the proof.
A little portion of algebraic group theory ( [2,12]). Let K be an algebraically closed field. Let K [X] be a polynomial ring in variables X1,... ,Xn over K. By an affine variety we mean a subset A C Kn which is a set of zeroes of some ideal in K [X] or equivalently it is of the form V(S) = {x e Kn : f (x) = = 0, Vf e S},where S is any set of polynomials in n variables over K. We consider the Zariski topology on Kn,n > 1 in which the closed sets are algebraic sets.
Let k be a subfield of K. An algebraic set is k-closed if it is V(S) for some S consisting of polynomials over k. The k-closed sets form a k-topology on Kn which is weaker than the original topology. If A is an arbitrary subset in Kn then the set I (A) consisting of all polynomials vanishing on A form an ideal — the annulator of A in K [X] . We say that an algebraic set A is k-defined (or a k-set) if I (A) can be generated by polynomials with coefficients in k. We shall also say that A is a k-variety, and denote by k[A] the algebra of regular functions defined over k. This is the quotient of the polynomials on Kn with coefficients in k by its subideal of polynomials vanishing on A. One can clearly speak of regular maps (over k) between k-varieties by examining the coordinate functions.
Recall that a field k is perfect if every irreducible polynomial over k has distinct roots. If k is perfect then an algebraic set A is k-defined iff A = V (S) and S consists of polynomials over k (see [12])).
An affine algebraic group G over K is an affine variety with group structure given by regular functions. Equivalently it is a nonsingular part of an algebraic set in Mn(K) — the set of n x n matrices over K. For a k—group G we let G(k) = G n GLn(k) be the set of k-rational points of G. We say also that G(k) is a k-portion of G.
An algebraic set A is irreducible if I (S) is a prime ideal. In terms of Zariski topology A is irreducible if it is not empty and is not the union of two proper closed subsets. The latter condition is equivalent to the requirement that each non-empty open set be dense in A, or that each one be connected. Every algebraic set can be decomposed into a finite union of irreducible subsets, called the components. In case A is an algebraic group, the notions of irreducibility and connectivity are equivalent. In what follows we use only the second term, to avoid confusion with the concept of irreducibility in the sense of representation theory.
Jordan-Chevalley decomposition. Recall that a matrix u e Mn (k) is called unipotent if (u — 1)n = 0. A matrix s e Mn (k) is semisimple if any s—invariant subspace in kn possesses an s—invariant complement subspace. In case of perfect k this is equivalent to diagonalizibility of s over the algebraic closure [12, § 15.2.1]. The notions just introduced do not depend on the choice of a perfect field k [12, § 15.1.5]. Also these notions are invariant under conjugation by matrices from GLn(k).
If k is perfect then every matrix x e GLn(k) has a unique Jordan-Chevalley decomposition x = xsxu where xs,xu e GLn(k), xs is semisimple, xu is unipotent and xsxu = xuxs [12, §15.3.3]. Moreover, xs,xu are the polynomials in x over k. We call the components of x semisimple and unipotent ones. By Mal'cev's theorem in every k-group G the components of any element belong to G(k) [12, §16.1.4].
Theorem 3. [12, § 18.1.4]. Let K be an algebraically closed extension of a perfect field k and G a commutative algebraic k-group. Then the semisimple and unipotent elements constitute the k-subgroups Gs and Gu respectively and G = Gs x Gu . The same decomposition is true for k—portions: G (k) = Gs (k) x x Gu (k).
The following properties are well known.
Lemma 2. Let k be a perfect field. (i) For any x,y e GLn (k) we have (yxy-1)s = yxsy-1 and (yxy-1)u = yxuy-1 , (ii) For any commuting a, b e GLn (k) the following holds true: (ab)s = asbs and (ab)u = aubu.
Proof. (i) We have yxy-1 = yxsxuy-1 = (yxsy-1) (yxuy-1). As yxsy-1, yxuy-1 commute and are semisimple and unipotent respectively, the desired equalities follow from the uniqueness of Jordan-Chevalley decomposition. (ii) Let G be an algebraic group generated by a,b, that is the intersection of all algebraic groups containing a, b. Then G is a commutative k-group (see [2, Ch.I, § 2.4] ) and the preceding Theorem shows that G = Gs x Gu. It follows that asas e Gs,aubu e Gu and thus ab = (asbs) (aubu) is a Jordan-Chevalley decomposition for ab. Whence the formulas. ■
Lemma 3. Cf. [26] Let k be a perfect field. Let x,y,z, in GLn(k) be such that
sys, 2)zn!
[x,y] = z and z commutes with both x and y. Then 1) xysx 1 = zsys, 2)zn! = 1.
In particular, the element is unipotent.
Proof. Rewrite the equality [x,y] = z in the form xyx-1 = zy. Taking the semisimple components of the last equality, and using the preceding Lemma 2, we obtain the formula xysx-1 = zsys. Iterating this formula, we get
xq ysx-q = z« ys (4)
for all natural q. A fact from linear algebra is that any commuting set S of diagonalizable linear endomorphisms of finite-dimensional vector space V can be simultaneously diagonalized over the algebraic closure k [5, Section 6.5., Theorem 8]. Then, taking the commuting semisimple elements zs and ys in diagonal form
Zs = diag {zi,... ,Zn} ,ys = diag {yi,... ,yn}
over the algebraic closure k (as we may), we see that the matrices z«ys (q > 0) are all diagonal and pairwise conjugated. Therefore, (4) implies zfy1 = yj(q) for some function q m j (q) e {1,...,n}. It follows that zf y1 = zf y1 for some distinct q1,q2 e [1; n + 1]. Hence z« = 1, where 1 < q < n and so zn! = 1. Similarly, z™! = 1 for all i. Hence zn! = 1 and zn! is unipotent. ■
2.1. The proof of Theorem 2
Let K be an algebraically closed extension of a perfect field k . By assumption, j — i > 2, hence there is a natural r such that j > r > i . The subgroup generated by xir (R) ,xrj (R) ,xj (R) is naturally isomorphic to UT3(R), therefore we may assume henceforth that m = 3 and xij- (R) = Z. To simplify notation, we assume that U = UT3(R) is contained in GLn(k). Of course, this embedding is assumed to be a monomorphism of abstract groups. In particular, apriori U may not be an algebraic subgroup in GLn(k) in any reasonable sense. We denote by G the Zariski closure of a group G < GLn(K) and we denote by G° the connected component of an algebraic group G < GLn(K). Recall that G° is a finite index subgroup in G and if G is k-defined then G° is an algebraic k-defined group [2, Ch.I, §.1.2].
By Mal'cev's theorem in an algebraic group G the components gs,gu of any element g belong to G and, moreover, if g is k-rational then gs,gu are k-rational too [12, § 16.1.4] . First, note that there is a dense open subset in Z consisting of the commutators. Indeed, Lemma 1 implies that the commutator map c : U x U m Z is surjective. It is obvious that the map c is regular, so its extension c : U x U m Z is also a regular map. We can therefore apply the Chevalley theorem [2, Theorem AG.10.2] to conclude that the image of c contains a dense open subset C in Z. It follows that the intersection C nZ is a dense open subset in Z consisting of commutators. By Theorem 3 we have Z = Zs x Zu . We assert that Z° = 1. Suppose not. Then setting Tn = |x e Z° : xn! = 1 j, we obtain a proper closed
subgroup Tnx Zu in a connected algebraic group Z° hence its complement D is a dense open subset in Z . The intersection of two open dense subsets in a connected variety is nonempty hence the intersection C n D is nonempty and open in Z° hence there is z = zs zu e C n D. By Lemma 3, the element zq is unipotent
for some q dividing n! . We conclude that zq = 1 and this implies that z G Tn x x ZM, contradicting the inclusion z g D. Finally, we have that Z = Zs x ZM and the connected component Z0 is trivial, hence Zs is finite and thus Z is virtually unipotent k—group.
3. Algebraic rings
Let K be an algebraically closed extension of a perfect field k. By an algebraic k-ring we mean an affine algebraic k-variety R over K with the k-regular maps "addition" and "multiplication", which turn R into a ring (possible non-associative and without identity). In particular this means that the additive group (R, +) is an affine abelian algebraic k-group. Note that for every r e R the left multiplication map R m R given by x m rx (x e R) is everywhere defined on R and hence it is regular, see [12, § 8.1.9]. Similarly, the right multiplication map is regular also.
Lemma 4. Cf. [8] Let R be an algebraic k-ring. Then the connected component R0 of an additive group (R, +) is a two-sided ideal in R. Moreover, R0 is an algebraic k—ring.
Proof. Recall that the connected component R0 is a subgroup of finite index in (R, +) whose cosets are connected, as well as irreducible, components of (R, +). Since R is defined over k, so is R0. For any r e R the map x m rx takes R0 onto rR0, hence rR0 is irreducible, see [12, § 8.2.5] . As it contains 0, it is contained in R0. This shows that R0 is a left ideal in R. Similarly, R0 is a right ideal. The last assertion follows directly from definitions. ■
Recall that a ring R (not necessarily associative) is an algebra over a field K if an operation K x R m R is defined such that this operation together with the addition constitute a structure of K—vector space and the following axioms are satisfied:
(Aa) b = a (Ab) = A (ab) for all A e K, a, b e R. (5)
Lemma 5. Let R be an algebraic k-ring over field K of characteristic zero. Suppose that the connected component R0 of the group (R, +) is a unipotent group. Then R0 has a structure of finite dimensional K—algebra which is compatible with the ring structure on R0. Moreover, R0(k) has a structure of a finite dimensional algebra over k which is compatible with the ring structure on R0(k).
Proof. By assumption (R, +) is a commutative unipotent group. Hence there is a k-isomorphism R0 ~ Kn. Thus we have a structure of a K-vector space on R0 (and the induced k-vector space structure on R0 (k) ). We are going to show that this structure is compatible with the ring structure on R0 in a sense that the axioms (5) hold true. Note that the axiom (Aa) b = A (ab) means that the operator rb of right
multiplication by b should be K-linear. But rb is a regular endomorphism of the group (Kn, +), hence it is a K-linear map (here the characteristic assumption is decisive). The second axiom can be verified similarly. Finally, the k-isomorphism R0 ~ implies that R0 (k) = kn and the k-algebra structure on kn is compatible with the ring structure on R0 (k). ■
Theorem 4. Let k be a field of an arbitrary characteristic and let R be a unital connected algebraic associative k-ring. Then the multiplicative group Rx of R is a linear k-group.
Proof. The group G = {(x,y) e R x R : xy = 1} is an affine algebraic k-group and thus Rx = G n (R, 1) is also an affine algebraic k-group. Every affine k-group is k-isomorphic to a closed subgroup, defined over k, of some GLn (K) (see [2, ch. I Prop. 1.10], and [12, § 31.23]), hence also is linear over k. ■
4. The proof of Theorem 1
We follow the scheme outlined in [8], overcoming some technical difficulties. Suppose that E3(R) is linear over k. To simplify the notation we assume that E3(R) is contained in GLn(k). Let Z denote x13 (R). The Zariski closure Z in GLn(K) is a commutative algebraic k-group. Henceforth we use an additive notation for this group operation. By Theorem 2, the group Z° is unipotent.
In order to define the multiplication on Z we need to use the following monomial matrices in E3(R) :
u
( o -1 o N 1 0 0
V
0 0 1
and v
100 0 0 1
V
0-10
/
They satisfy the following key properties (we denote conjugation by ux
uxu
-1):
ux13(r) = x23 (r) and v x^(r) = x^(r).
Recall again that we consider E3(R) as a subgroup in GLn(K). The conjugacy operations g mu g,g m vg are k—regular maps on GLn(K). Let us define a k-regular map ■ from Z x Z to GLn(K) as follows:
x ■ y := [vx,u y] (x,y e Z) .
Since ■ takes Z x Z into Z, it also takes Z x Z into Z ( [2, ch. I, sec.6.6.]). Moreover, the commutator relation [x12(r),x23(s)] = x13(rs), (r, s e R) implies that the restriction of ■ to R c Z coincides with the original multiplication on R. The element 1 e R is a unit in R hence it is unit (with respect to ■) in its closure R = Z . Similarly, since R is associative, R = Z is associative also. We have introduced an associative unital ring structure on Z D R extending the original ring structure on R. By Lemma 5, the group (k) is a two-sided ideal in Z(k) and (k) is linear over k hence Z0 (k) n R is a two-sided ideal in R which is linear over k. The proof of Theorem 1 is complete.
5. Nonlinear Witt rings R with linear GLn (R)
The analog of Theorem 4 is not true in the case of positive characteristic. Here is one simple example given in [8] (it is somewhat similar to the example from [1] ): Let K be an infinite field of characteristic 2 and let us give R = K x K the following operations:
(a, b) + (c, d) = (a + c, ac + b + d), (a, b) x (c, d) = (ac, bc2 + a2d). (6)
One can verify directly that R becomes a commutative local ring with the maximal ideal M = (0,K) and residue field R/M ~ K. Therefore R does not have proper ideals of finite index. This ring is not linear over any field since all elements of the form (a,b), a = 0, have "additive" order 4. Hence R is not virtually linear. Rather surprisingly, GLn (R) is linear for all n > 2 ! Thus, there exists a (strange) ring R which is not (virtually) linear over any field, but the group ELn(R) is linear ! In this section we show that the all truncated Witt rings Wn (k) over infinite perfect field of finite characteristic possess this property.
5.1. Witt rings are strange
Our basic reference for this section is Serre [Ser] (see also [7,24]). From now on, we let p be a fixed prime number. For n > 0 define the n-th Witt polynomial to be
n
Wn = ^piXfe Z[Xo, X1,..., Xn]. (7)
i=0
Thus
Wo = Xo
Wi = XP + pXi
W2 = xp2 + pXP + p2X2
Wn = XoP + pXP + ••• + pnXn. If we extend the coefficient ring to Z 1 or even to a larger ring A, then the
equations can be inverted:
Xo = Wo ,Xi = p-1 (Wi - WQ ),
, etc.
Theorem 5. 7%ere ex/si unique sequences (S0,..., Sn,...), (P0, polynomials in Z [X0,..., Xn,...; Y0,..., Yn,...], such that:
Wn (S0,..., Sn) = Wn (X0, ...,Xn) + Wn (Y0,..., Yn),
Wn (P0, . . . , Pn) = Wn (X0, . . . , Xn) Wn (Y0, . . . , Yn) for all n = 0, 1 . . ..
(8)
Pn,...) of
Example 1.
X + YP - (Xo + Yo)P
S0 = X0 + Y0, S1 = X1 + Y1 +
-
P0 = X0Y0, P1 = Y + X1YP + pX1Y1.
Let A be a unital commutative ring. We have the usual, product ring structure on An+1. Define a new addition and multiplication in An+1 by
a+b = (S0 (a, b),..., Sn (a, b)),
a ■ b = (P0 (a, b) ,...,Pn (a, b)).
These laws of composition make An+1 into a commutative unital ring, called the ring of (truncated) Witt vectors and denoted Wn (A). The zero and unit elements of Wn (A) are 0 = (0,..., 0) and 1 = (1,0,..., 0) respectively. In case n = 1 we have a ring isomorphism W0 (A) ~ A. The map
W* : Wn (A) m An+1
which assigns to a Witt vector a = (a0, . . . , an) the element
(W0 (a0) , W1 (a0, a1),..., Wn (a0,..., an))
of the product ring An+1, is a ring homomorphism by the very definition of the polynomials S and P. It follows also that the projection map (a0,... ,an) m a0 is a ring homomorphism from Wn (A) to A.
Example 2. The ring structure W1 (A) is defined on A2 according formulas (8), so in coordinates x = (x0,x1), y = (y0,y1) it looks as follows:
(x+y)0 = x0 + y°, (x+ y) 1 = x1 + y1 + - (x0 + yp — (x0 + y0)p),
(x ■ y)0 = XoУo, (x ■ y)1 = xpy1 + x1yp + -x1y1.
In case p =2 we obtain
(x+y) 0 = xo + yo, (x+y) 1 = xi + yi + xoyo, (x ■ y)o = xoУo, (x ■ y)i = x^yi + xr^
which coincides with the Kassabov-Sapir structure given by 6.
Proposition 1. Let K be any algebraically closed field and k its prime sub-field. The Witt ring Wn (K) is a unital commutative algebraic k-ring. Moreover, the set of k-rational points is canonically isomorphic to the ring Wn (k).
Proof. The ring Wn (K) is algebraic because the underlying k-variety is Kn+1 and ring operations are polynomial over the prime subfield. The k-points are kn+1 c Kn+1 and the operations are given by the same polynomials as in the definition of Wn (k). ■
"Verschiebung" V and Frobenius F. One defines the Verschiebung (=shift) map V : Wn(A) ^ Wn(A) by
(X0 ,X1,...,Xn) ^ (0, X0, ..., Xn-1).
This map is additive (in Witt ring structure). In case A is the ring of characteristic p the Frobenius map F : Wn(A) ^ Wn(A) is defined by
(x0, x1, ..., xn) ^ (x0, X1, ..., Xn).
These maps satisfy identities VF = FV = p , where p denotes the p-power map on the additive group of Wn(k).
Theorem 6. Let A be an infinite integral domain of characteristic p > 0. The ring W0(A) is linear over the quotient field K of A. For every natural n the Witt ring Wn(A) is not virtually linear over any field.
Proof. As we know, W0(A) ~ A, and A is clearly linear over K. When n > 1 we first show that Wn (k) has an additive exponent pn+1 and, moreover, all Witt vectors (x0, ...,xn) with x0 = 0 have an additive order pn+1. Iterating the formula VF = FV = p, we obtain (VF)k = (FV)k = pk for any natural k. Hence
/ k k \ p (x0,... ,Xn) = (^0,..., 0,xp ,... ,xn_fcJ ,
from which it follows that pn+1x = 0 for each x e Wn(A). Moreover, if x0 = 0 then pn (x0,..., xn) = (0,... 0, x0") = 0.
Finalizing the proof, let I be an ideal of finite index in Wn (A), which is linear over some field L. The ideal I can not lie entirely in the ideal M = (0, A,..., A), since Wn (A) /M ~ k is infinite by assumption. Hence, there is x = (x0,..., xn) e I with x0 = 0 but then pn+1x = 0 and pnx = 0, which can not happen in an L-algebra. ■
Theorem 7. For any field k and any m, n > 1 the group GLm (Wn (k)) is linear over k.
Proof. Since Wn (k) is algebraic, the full matrix ring R = Mm (Wn (k)) is an algebraic k-ring. By theorem 4 the multiplicative group Rx is linear over k. Hence its subgroup GLm (Wn (k)) is linear over k also. ■
6. Questions
1. It is proved in [8] that any ring homomorphism of a free associative ring Z(x,y) into an algebraic ring R has a non-trivial kernel. Is it possible to embed Fp(x, y) into an algebraic ring over a field of positive characteristic? (see Remark 16 in [8]).
2. What is the structure of affine (non-commutative) algebraic rings?
3. What is the structure of (not necessarily affine) algebraic rings? The complement by George M. Bergman to the book [18] hopefully might be helpful.
4. Let R be an almost linear ring over a field k. Is it true that En (R) is a linear group over k ?
Acknowledgements
This research was partially supported by DFG through SFB 701 of Bielefeld University and by RFFI-grant 14-01-00068.
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НЕОБХОДИМОЕ УСЛОВИЕ ЛИНЕЙНОСТИ НАД ПОЛЕМ ДЛЯ ЭЛЕМЕНТАРНОЙ МАТРИЧНОЙ ГРУППЫ
Г.А. Носков
д.ф.-м.н., с.н.с., e-mail: g.noskov@googlemail.com
Институт Математики им. С.Л. Соболева СОРАН
Аннотация. Мы доказываем, что если R есть ассоциативное кольцо с единицей, и элементарная матричная группа En(R) при n > 3 линейна над полем k нулевой характеристики, то в R имеется идеал конечного индекса, линейный над k. Доказывается, что, если A является коммутативным целостным кольцом ненулевой характеристики, то для любого натурального n кольцо векторов Витта Wn(A) не является почти линейным ни над каким полем. В то же время, несколько парадоксально, общая линейная группа GLm(Wn(k)) линейна над k в случае произвольного поля k.
Ключевые слова: линейная группа, поле, аффинная алгебраическая группа, ассоциативное кольцо, кольцо Витта.
