communications . In
COMMUNICATIONS
IN ALGEBRA, 30(1),489-505 (2002)
GRAD ED IDENTITIES OF GROUP ALGEBRAS S. K. Sehgal) and M. V. Zaicev2 IDepartment of Mathematical Sciences, University of Alberta, Edmonton, AB T6G 2G 1, Canada E-mail:
[email protected] 2Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119899 Russia E-mail:
[email protected]
ABSTRACT Let H be a normal subgroup of G. Then the group algebra A = FG can be naturally graded by G/ H where the homogeneous components are cosets. We prove that if A satisfies a G/ H-graded identity than it also satisfies an ordinary polynomial identity under the assumption that [G : H] is finite.
1 INTRODUCTION Let FG = A be the group algebra of a group G over a field F. The algebra A can be naturally graded both by G itself and by any quotientgroup G/ H. Recall that an associative algebra is said to be P.I. if it satisfies a non-trivial polynomial identity. The classification of groups whose group algebra is P.I. is well-known (see [1], Chapter 5). We investigate the relation 489 Copyright ~ 2002 by Marcel Dekker, Inc.
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between graded and non-graded identities on A. Namely, we are interested in knowing if an arbitrary graded identity on A implies a usual P.I. If H has infinite index in G then it is possible for FG to be a G/ H-graded P.I. but not P.I. A trivial counterexample is given by free non-abelian group G with H = {e}. Its group algebra FG = tBgEGFgis not P.I. but the neutral component Fe is a commutative subalgebra and xy = yx Vx,y E Fe is a graded identity on FG. So we are interested in the case where G/ H is a finite group. For an arbitrary associative algebra B graded by a finite group the existence of a graded identity on B is not enough for B to be P.I. For example, a free associative algebra A = F(X) admits a Z2 grading, A = Ao tBA I, where Ao is the span of all monomials in X of even degree, and A I is the span of monimials
of odd degree. But the grading group H
= Z2
can be imbedded in some other finite group G and A can be considered as a G-graded algebra with Ag = 0 for any g E G \ H. Obviously, A satisfies Ggraded identity x ==0, x E Ag, g E G \ H. The main result of the present paper, Theorem 2, shows that if FG satisfies a G/ H-graded identity g - 0 with G/ H finite then it also satisfies some ordinary polynomial identity f - 0 and degf is restricted by some function of deg g and [G : H]. Sections 2 and 3 contain necessary definitions and notations. In Sec. 4 we prove a technical result, Theorem I, which gives us a sufficient condition. In Sec. 5 we reduce the general problem to Theorem 1.
2 GRADED ALGEBRAS AND GRADED IDENTITIES Let G be an arbitrary group. An associative algebra A over F is said to be G-graded if A is a direct sum of its subspaces Ag, g E G, and AgAIl ~ Agll. To define a graded identity on A we need some additional construction. Let X = UgEGXg be a disjoint union of its subsets Xg, g E G, where Xg = {xf, 4, . . .}. Consider a free associative algebra F(X) generated by X. This algebra admits a natural G-grading, F(X) = gEG tB F(X) g if we put xf.1...x7,;' E F(X)g if and only if g,...gll
=g.
We say that a
is a homomorphism of graded algebras (or graded homomorphism) if q>(F(X)g) ~ Ag for any g E G. Obviously, any map p : X 1--* A such that p(Xg) ~ Ag for any g E G can be extended to a homomorphism F(X) 1--* A of graded algebras. A polynomialf = f(xf',. . . ,X~II)defines a graded identity on A if flies homomorphism
q>: F(X)
1--* A
in the kernels of all graded homomorphisms F(X) 1--*A. In other words,
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GRADED IDENTITIES OF GROUP ALGEBRAS
f = f(xr, . . . , x~,,) = 0 is a graded identity of algebra A if .f(al,. .. , an) = 0 for any al E Ag". . . , all E Ag". If an algebra A satisfies an ordinary polynomial identity .f(xl,.. ., XII)
- 0 then it also satisfies a graded identity f = f(xfl , . .. ,x~,,)= 0 for any gl,... ,gll E G. Hence any P.I. algebra which is graded by some group G satisfies also some graded identity but not conversely. The most simple counterexample is given by an arbitrary algebra B without polynomial identities and with the grading B = B" Bg = 0 for any I -=1=g E G, G being any non-trivial group. This algebra satisfies all identities of the type xg - O,g E G,g -=1= l. The theory of ordinary P.I. algebras is elaborated very well, see, for instance, [2, 3]. The example given above shows that one cannon extend all results from ordinary P.I. algebras to graded P.I. algebras.
3
GRADINGS ON GROUP ALGEBRAS
---
Now let G be an arbitrary group and A = FG be its group algebra.
Then A is the F-linear span of all g E G. If we denote Ag = (g)F then A
= gEG EB Ag,
AgA"
= Ag",
i.e., A is naturally graded by G. Consider a normal subgroup H in G and the quotient group T = G/ H. If we define At =
EB Ag, gH=t
t E T,
then we get a T-grading on A since A = EBtETAt and AtAs Yk2+1,Yk2+2,...,Yk",...,Yk".1 >Yk,,_,+I,...,Ykr Now let aO=Yl...Yk,-I,
ad+1 = Ykd+1 .. .YI1,
and a, = Yk,Yk,+1.. .Yk!fI-l for I = I,... ,d. In case k1 = 1 or kd = n we suppose that ao (resp. ad+l) is 1 E F. Now we can write Iz-I
E= (a001)(al...ad0gi,...gi,,)(ad+10gi,,+,
...g;J
where h = gil . . .gi" E P and gi,,+1.. . gi" = 1 if d = n. But from relation (6) (or inclusion (7ยป follows that intermediate factor a( . . . ad 0 gh . . "gi" modulo I .. " " ifl if! " -1- 1 1 b f h S IS a mear
com
IllatIon
0
some
aa(l) . . . aa'(d) 0 t( . . . td WIt
UI
,u E
d.
Note that a~(I) < al if u(l) i= 1 since a~(1)starts with Yt\l) which is less than Yk,. Hence the product aoa~(I). . . a~'(d)d is lexicographically less than Goal. .. ad+1for any monomial d. The same conclusion holds for any u E Sd such that u(1) = 1,u(2) > 2, or u(l) = 1,u(2) = 2,u(3) > 3, and so on. Since I is two-sided ideal in F(XIP, tjI), we get an expression of E as the sum of elements from I and linear combination of monomials of the form ifl
if"
aOaa(I)...aa(d)ad+l
tl...t"'I.IrQ-,
t
' 0 one has btf(n) < ell n! as soon as n > p(d, e) where btf(n) is the number of d-indecomposable multilinear words in n letters. It follows that for any fixed p =
SEHGAL AND ZAICEV
498 (PI, . . . ,Pn) E pil and for any g E P the number monomials of the type
of d-indecomposable
l/!(p"(I)) I/!(P"(II)) @ XIT(I) ...XIT(Il) g
is less than ~. Hence the total number of d-indecomposable monomials in VIl,gis less than ~r1 since group Q = tjJ(P) has order r. But from (11) and (12) follows that number of d-indecomposable monomials in WI1is less than I'llmil. Therefore if we take, for instance, c = I'm + I then by Lemma I the dimension Wil modulo I is strictly less than n! for any n > p(d, rm + I). For
~
no = p(d, rm + I) + I the inclusion (8) holds as it was mentioned
above and
B = AI = FH satisfies a non-trivial polynomial identity of degree no. To finish the proof of Theorem I note that A = FG is aT-graded algebra with polynomial identity of degree no on the neutral component AI and T is a finite group of order m. It was shown in [6] that A itself has an identity of degree bounded by some function depending only on no and m. Remark. The final conclusion may also be obtained from earlier classification of P.I. group algebras. Indeed, from [7, 8] (see also [I], Theorems 5.3.7 and 5.3.9) follows that H contains a subgroup D of finite index. In addition its commutator D' is trivial (i.e., D is abelian) if char F = 0 or D' is a finite p-group if char F = P > O. Anyway, [H: D]ID'I is restricted by some function on no- But in this case D is a subgroup of G with the same properties and [G: D]ID'I = m[H: D]ID'j, hence FG is also P.I. and minimal degree of an identity on FG is restricted by a function in no and m. 0 From Theorem 1 immediately follows that if G is a split extention of a group H by a finite group T ~ G / H then any T-graded identity on FG implies an ordinary one.
5
MAIN RESULT
Theorem 2. Let H be a normal subgroup of finite index m in a group G. Consider the natural G/ H-grading on the group algebra FG: FG = A =
EB At. tEG/H
If FG satisfies a graded identity of degree d then it also satisfies an ordinary polynomial identity of degree n restricted by some function of d and m.
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The last Theorem was proved by purely combinatorial methods. In order to prove our result in the general case we need to use also some other arguments. Next two lemmas are generalization of some technical results from [8] (see also [I], Lemma 5.2.4 and Theorem 5.2.6) to the case of graded identities. Denote by
