An uncountable family of almost nilpotent varieties of polynomial growth

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by  S. Mishchenko

AN UNCOUNTABLE FAMILY OF ALMOST NILPOTENT VARIETIES
OF POLYNOMIAL GROWTH
S.MISHCHENKOANDA.VALENTI
7
1
0 Abstract. Anon-nilpotent varietyofalgebrasisalmostnilpotentifanypropersubva-
2 rietyisnilpotent. Letthebasefieldbeofcharacteristiczero. Ithasbeenshownthatfor
associativeorLiealgebrasonlyonesuchvarietyexists. Herewepresentinfinitefamilies
n
ofsuchvarieties. Morepreciselyweshallprovetheexistenceof
a
1)acountablefamilyofalmostnilpotentvarietiesofatmostlineargrowthand
J
2)anuncountable familyofalmostnilpotentvarietiesofatmostquadraticgrowth.
3
2

]
A
1. Introduction
R
. Let F be a field of characteristic zero and F{X} the free non associative algebra on a
h
countable set X over F. If V is a variety of non necessarily associative algebras and Id(V)
t
a is the T-ideal of polynomial identities of V, then F{X}/Id(V) is the relatively free algebra
m
ofcountablerankofthevarietyV. Itiswellknownthatincharacteristiczeroeveryidentity
[ is equivalent to a system of multilinear ones, and an important invariant is provided by the

sequence of dimensions c (V) of the n-multilinear part of F{X}/Id(V), n =1,2,.... More
1 n
v precisely, for every n ≥ 1 let Pn be the space of multilinear polynomials in the variables
9 x ,...,x . Since char F = 0, F{X}/Id(V) is determined by the sequence of subspaces
1 n
6 {P /(P ∩ Id(V))} and the integer c (V) = dimP /(P ∩ Id(V)) is called the n-th
n n n≥1 n n n
4
codimension of V. The growthfunction determined by the sequence of integers {c (V)}
6 n n≥1
is the growth of the variety V.
0
. In general a variety V has overexponential growth, i.e., the sequence of codimensions
1
cannot be bounded by any exponential function. Recall that V has exponential growth if
0
7 cn(V)≤an,foralln≥1,forsomeconstanta. Forinstanceanyvarietygeneratedbyafinite
1 dimensionalalgebrahasexponentialgrowth. Forsuchvarietiesthelimitlimn→∞ pn cn(V)=
: exp(V), is called the PI-exponent of the variety V, provided it exists.
v
i We saythat a varietyV haspolynomialgrowthifthere existconstants α,t≥0suchthat
X
asymptotically c (V) ≃ αnt. When t = 1 we speak of linear growth and when t = 2, of
n
r quadratic growth.
a
MoreoverV has intermediate growthif for any k >0, a>1 there exist constants C ,C ,
1 2
such that for any n the inequalities
C nk <c (V)<C an
1 n 2
hold. Finally wesaythata varietyV hassubexponentialgrowthif foranyconstantB there
exists n such that for all n>n , c (V)<Bn. Clearly varieties with polynomial growthor
0 0 n
intermediategrowthhavesubexponentialgrowthanditcanbeshownthatvarietiesrealizing
eachgrowthcanbe constructed. Forinstanceaclassofvarietiesofintermediategrowthwas
constructed in [5].
2000 Mathematics Subject Classification. Primary 17A50, 16R10, 16P90; Secondary 20C30 Keywords:
polynomialidentity,cocharacter, codimension.
1
2 MISHCHENKOANDA.VALENTI
The purpose of this note is the study of the almost nilpotent varieties. Recall that a
variety V is almost nilpotent if it is not nilpotent but all proper subvarieties are nilpotent.
Aboutpreviousresults,ifweconsidervarietiesofassociativealgebras,itiseasilyseenthat
the only almostnilpotent variety is the varietyV of commutative algebras,(the sequence of
codimensionsisc (V)=1,n≥1). IncaseofvarietiesofLiealgebrasithasbeenshownthat
n
there is also only one almost nilpotent variety: the variety A2 of metabelian Lie algebras
and in this case c (A2) = n −1. In [3] it was proved that there exist only two almost
n
nilpotent varieties of Leibniz algebras and both varieties have at most linear growth. For
general non associative algebras, in [11] an almost nilpotent variety of exponent two was
constructed. Later in [10] it was proved that for any integer m an almost nilpotent variety
with exponent m exists. Recently in [8] it was proved the existence of almost nilpotent
varieties with fractional exponent.
An algebra satisfying the identity x(yz)≡0 will be called left nilpotent of index two. In
[12] two almost nilpotent varieties with linear growth were constructed and it was proved
thattheyrepresentafulllistofalmostnilpotentvarietieswithsubexponentialgrowthinthe
classofleftnilpotentalgebrasofindextwo. Forcommutative(anticommutative)metabelian
algebras similar result were obtained in [1], [9].
The purpose of this note is to prove the existence of two families of almost nilpotent
varieties. The first one is a countable family of at most linear growthand the second one is
an uncountable family of at most quadratic growth.
2. The general setting
ThroughoutAwillbeanonnecessarilyassociativealgebraoverafieldF ofcharacteristic
zero and F{X} the free non associative algebra on a countable set X = {x ,x ,...}. The
1 2
polynomial identities satisfied by A form a T-ideal Id(A) of F{X} and by the standard
multilinearization process, we consider only the multilinear polynomials lying in Id(A). To
this end,foreveryn≥1,wesetP tobe the spaceofmultilinearpolynomialsinx ,...,x ,
n 1 n
andweletthesymmetricgroupS actonP besettingσf(x ,...,x )=f(x ,...,x ),
n n 1 n σ(1) σ(n)
for σ ∈S ,f ∈P .
n n
The space P (A) = P /(P ∩Id(A)) has an induced structure of S -module and we let
n n n n
χ (A) be its character,called the n-th cocharacter of A. By complete reducibility we write
n
χn(A)=Xmλχλ
λ⊢n
whereχ isthe irreducibleS -charactercorrespondingtothe partitionλ⊢n andm ≥0is
λ n λ
the corresponding multiplicity (we refer the reader to [6] for an account of this approach).
We nextrecallsomebasicpropertiesofthe representationtheoryofthesymmetricgroup
that we shall use in the sequel. Let λ ⊢ n and let T be a Young tableau of shape
λ
λ ⊢ n. We denote by e the corresponding essential idempotent of the group algebra
Tλ
FS . Recall that e = R+ C− where R+ = σ, andC− = (sgnτ)τ
n Tλ Tλ Tλ Tλ Pσ∈RTλ Tλ Pτ∈CTλ
and R , C are the groups of row and column stabilizers of T , respectively. Recall
Tλ Tλ λ
that if M is an irreducible S -submodule of P (A) corresponding to λ, there exists a
λ n n
polynomial f(x ,...,x ) ∈ P and a tableau T such that e f(x ,...,x ) 6∈ Id(A). Let
1 n n λ Tλ 1 n
e′ = C−R+ C− . Since R+ C− R+ C− 6= 0 then e′ is a nonzero essential idempotent
Tλ Tλ Tλ Tλ Tλ Tλ Tλ Tλ Tλ
that generates the same irreducible module and so also e′ f(x ,...,x )6∈Id(A).
Tλ 1 n
In what follows we shall also denote by g(λ) the polynomial obtained from the essential
idempotent corresponding to a tableau of shape λ by identifying the elements in each row.
Recall that g(λ) is an highest weight vector of the general linear group GL (F) where k is
k
the number of distinct part of λ (see [2])
AN UNCOUNTABLE FAMILY OF ALMOST NILPOTENT VARIETIES OF POLYNOMIAL GROWTH 3
Now, for a fixed arrangement of the parentheses T, let us denote by PT the subspace
n
of P spanned by the monomials whose arrangement of the parentheses is T. Let also
n
PT(A)=PT/(PT ∩Id(A)). Then clearly P (A)= PT(A).
n n n n PT n
Since the S -module PT(A) is a homomorphic image of PT ≡ FS , the regular S -
n n n n n
representation, it follows that, if χ (A)T is the S -character of PT(A), then
n n n
χn(A)T =XmTλχλ
λ⊢n
and mT ≤d =degχ . Clearly m ≤ mT.
λ λ λ λ PT λ
Throughout we shall also use the following convention: we shall write the same symbol
(e.g. ¯, ˜) over two or more variables of a polynomial to indicate that the polynomial is
alternating on these variables.
For instance x x x =x x x −x x x .
3 1 2 3 1 2 3 2 1
We also need to recall some results from the theory of infinite words (see [7]). Recall
that, givenan infinite (associative)wordw in the alphabet{0,1}the complexity Comp of
w
w is defined as the function Comp : N → N, where Comp (n) is the number of distinct
w w
subwords of w of length n.
Also,aninfinitewordw =w w ··· isperiodicwithperiodT ifw =w fori=1,2,....
1 2 i i+T
It is easy to see that for any such word Comp (n) ≤ T. Moreover, an infinite word w is
w
called a Sturmian word if Comp (n)=n+1 for all n≥1.
w
Forafinite wordx,theheighth(x)ofxisthenumberofletters1appearinginx. Also,if
|x| denotes the length of the word x, the slope of x is defined as π(x)= h(x). In some cases
|x|
this definition can be extended to infinite words as follows. Let w be some infinite word
andlet w(1,n) denote its prefix subwordoflengthn. If the sequence h(w(1,n)) convergesfor
n
n→∞ and the limit
h(w(1,n))
π(w)= lim
n→∞ n
exists then π(w) it is called the slope of w. Examples of infinite words for which the slope
is not defined can be given. Nevertheless for periodic and Sturmian words the slope is well
defined. In the next proposition we reassume the main properties of these words that we
shall use here.
Theorem 1. ([7, Section 2.2]) Let w be a Sturmian or periodic word. Then there exists a
constant C such that
1) |h(x)−h(y)|≤C, for any finite subwords x,y of w with |x|=|y|;
2) the slope π(w) of w exists;
3) |π(u)−π(w)|≤ C , for any non-empty subword u of w;
|u|
4) for any real number α ∈ (0,1) there exists a word w with π(w) = α and w is
Sturmian or periodic according as α is irrational or rational, respectively.
If w is Sturmian we can take C =1, and if w is periodic of period t, then π(w)= h(w(1,t)).
t
3. Algebras constructed from periodic or Sturmian words
Our aim in this section is to prove the existence of two families of almost nilpotent
varieties. The firstisacountablefamilyofvarietiesofatmostlineargrowthandthe second
is an uncountable family of at most quadratic growth. To do this we will make use of an
algebra constructed in [4].
ThroughoutA will be the algebrageneratedby one element a such that every wordin A
containing two or more subwords equal to a2 must be zero.
4 MISHCHENKOANDA.VALENTI
Note that in particular the algebra A is metabelian, i.e., it satisfies the identity
(x x )(x x )≡0.
1 2 3 3
A partial decomposition of the cocharacter of A was given in [4] and we recall it here.
Let L and R denote the linear transformationson A of left andright multiplication by
a a
a, respectively. We shall usually write bL =L (b)=ab and bR =R (b)=ba.
a a a a
We have the following
Remark 1.
1) χ (A)=m χ +m χ
n (n) (n) (n−1,1) (n−1,1)
2) c (A)≥2n−2.
n
Proof. Let λ = (λ ,λ ,...) ⊢ n be a partition of n such that n−λ ≥ 2. This says that
1 2 1
either the first column of λ has at least three boxes or the first two columns of λ have at
least two boxes each. Hence, if f is an highest weight vector associated to λ, either f is
λ
alternating on three variables or f is alternating on two distinct pairs of variables. In both
caseseverymonomialoff evaluatedinAcontainsatleasttwosubwordsequaltoa2. Hence
λ
f ∈Id(A) and this implies that χ appears with zero multiplicity in the decomposition of
λ λ
χ (A). It follows tha
n
χ (A)=m χ +m χ
n (n) (n) (n−1,1) (n−1,1)
is the decomposition of χ (A) into irreducibles.
n
In order to prove 2) we compute the multiplicity m in χ (A).
(n) n
Let w(L ,R )∈End(A) be a wordin L and R of length n−2. Clearly a2v(L ,R )=
a a a a a a
v(L ,R )(a2) is the evaluation of an highest weight vector associated to the partition (n)
a a
whichis notanidentity ofA. Since there are2n−2 distinct suchwords,we get2n−2 highest
weight vectors which are linearly independent mod Id(A). Thus since degχ = 1, from
(n)
χ (A)=m χ +m χ , we have that c (A)≥2n−2. (cid:3)
n (n) (n) (n−1,1) (n−1,1) n
Next we shall compute the decomposition of the cocharacter χT(A) for a fixed arrange-
n
ment T of the parentheses of P .
n
We have the following
Proposition 1. For any arrangement T of the parentheses in P we have
n
(1) χ (A)T =χ +2χ .
n (n) (n−1,1)
Proof. If PT(A)6=0 then any monomial of PT is of the form
n n
x x T ...T (mod Id(A)),
σ(1) σ(2) 1,xσ(3) n−2,xσ(n)
where T =L or T =R , for any i,j.
j,xi xi j,xi xi
Itfollowsthat,modId(A),thehighestweightvectorscorrespondingtostandardtableaux
of shape (n−1,1) are
g (x ,x )=(x x )T ...T
0 1 2 1 2 x1 x1
and
g (x ,x )=(x x )T ...T T T ...T , 1≤i≤n−2.
i 1 2 1 1 1,x1 i−1,x1 i,x2 i+1,x1 n−2,x1
Recall that the symbol ¯over two or more variables of a polynomial means that the
polynomial is alternating on these variables.
We claim that for any 1≤i,j ≤n−2 the elements g (x ,x ) and g (x ,x ) are linearly
i 1 2 j 1 2
dependent mod Id(A). In fact, since any word containing two subwords equal to a2 is zero
in A, in a non-zero evaluation ϕ we must set ϕ(x )=a and ϕ(x )=a2v(L ,R ), for some
1 2 a a
monomial v(L ,R )∈End(A).
a a
AN UNCOUNTABLE FAMILY OF ALMOST NILPOTENT VARIETIES OF POLYNOMIAL GROWTH 5
We get
ϕ(g (x ,x ))=ϕ(g (x ,x ))=−a2v(L ,R )R T ...T ,
i 1 2 j 1 2 a a a 1,a n−2,a
and the claim is established.
Next our aimis to provethat the polynomialsg (x ,x ) andg (x ,x ) are linearlyinde-
0 1 2 1 1 2
pendent mod Id(A). In fact suppose that αg (x ,x )+βg (x ,x ) is an identity of A, for
0 1 2 1 1 2
some α,β ∈F. If we consider the evaluation ϕ(x )=a and ϕ(x )=a2, we get
1 2
αg (a,a2)+βg (a,a2)=αa2L T ...T −(α+β)a2R T ...T ,
0 1 a 1,a n−2,a a 1,a n−2,a
and the right hand side is zero only if α=β =0.
Wehaveprovedthatχ appearswithmultiplicity2inthedecompositionofχ (A)T.
(n−1,1) n
Since mT =1 we get that χ (A)T =χ +2χ and the proposition is proved. (cid:3)
(n) n (n) (n−1,1)
Next for every real number between 0 and 1 we shall construct a quotient algebras of A.
To this end we keep in mind the terminology of the previous section.
We are going to associate to every finite word in the alphabet {0,1} a monomial in
End(A) in left and right multiplications: if u(0,1) is such a word we associate to u the
monomial u(L ,R ) obtained by substituting 0 with L and 1 with R .
a a a a
Let α be a real number, 0 < α < 1, and let w be a Sturmian or periodic infinite word
α
in the alphabet {0,1} whose slope is π(w )=α.
α
Let I be the idealofthe algebraA generatedby the elements a2u(L ,R ) where u(0,1)
α a a
is not a subword of the word w .
α
Let A = A/I denote the corresponding quotient algebra and let V be the variety
α α α
generated by the algebra A .
α
We have
Lemma 1. For anyreal numberα, 0<α<1, the variety V has linear or quadratic growth
α
according as w is a periodic or a Sturmian word.
α
Proof. We are going to find an upper and a lowerbound of the codimensions of the algebra
A . To this end we start from the decomposition of the cocharacter of A given in (1).
α
Letn≥3 be anyintegerandletu(0,1)be a subwordofthe wordw oflengthn−1. We
α
may clearly assume that 0 is the leftmost symbol of such word and, so, we write u(0,1) =
0v(0,1) for some subword v(0,1) of w .
α
Since u and v are subwords of w , a2v(L ,R ),a2L v(L ,R ) 6∈ I . This implies that
α a a a a a α
the polynomial
g (x ,x )=(x x )v(L ,R )
0 1 2 1 2 x1 x1
isnotanidentityofthealgebraA . Infact,recallthattheevaluationϕ(x )=a,ϕ(x )=a2
α 1 2
gives ϕ(g (x ,x ))=a2L v(L ,R )−a2R v(L ,R )6∈I .
0 1 2 a a a a a a α
Since the wordu(0,1)wasanarbitrarysubwordofthe wordw oflengthn−1,this says
α
that, for any corresponding arrangementT of the parentheses in
(2) χ (A )T =χ +mT χ
n α n (n−1,1) n−1,1
wemusthavemT >0.Moreovercomparethelastequalitywith(1)andrecallthat,since
(n−1,1)
A isaquotientalgebraofA,themultiplicities inχT(A )areboundedbythe multiplicities
α n α
in χT(A). It follows that 0<mT ≤2.
n (n−1,1)
Now, the different arrangements of the parentheses in nonzero words of length n in A
α
correspond to the subwords of w of length n. Recalling that Comp (n) = constant or
α wα
Comp (n) = n+1 according as w is periodic or Sturmian, respectively it follows that
wα α
their number is bounded by a constant in case α is rational (i.e., w is periodic) and by a
α
linear function of n in case α is irrational (i.e., w is Sturmian).
α
6 MISHCHENKOANDA.VALENTI
Since degχ = 1 and degχ = n−1, from (2) and the above discussion we can
(n) (n−1,1)
find constants C ,C such that for any n we have
1 2
C n≤c (A )≤C n,
1 n α 2
if α is rational, and
C n2 ≤c (A )≤C n2
1 n α 2
if α is irrational.
Recalling that the growth of V is the growth of the sequence c (A ) the proof of the
α n α
lemma is complete. (cid:3)
Proposition 2. For 0<α<β <1, the variety V ∩V is nilpotent.
α β
Proof. Let K (w ) denote the set of different subwords of length n of a word w . now, the
n γ γ
slope of the words w and w is equal to α and β, respectively. Since α6=β, by Theorem 1
α β
there exist m such that for any n≥m the intersection K (w )∩K (w ) is the empty set.
n α n β
In particular there exist m such that any word u(0,1) of length m is not a subword either
of the word w or of the word w .
α β
Let for instance u(0,1) be a word of length m which is not a subword of the word
w , and consider the monomial y y u(L ,R ). Construct the multilinear element y y u
α 1 2 x x 1 2
on y ,y ,x ,...,x where u is obtained by substituting x ,...,x instead of x inside
1 2 1 m 1 m
u(L ,R ). Hence y y u≡0is anidentity ofthe varietyV . Itfollowsthat y y u≡0is also
x x 1 2 α 1 2
anidentityofV ∩V ,andsoc (V ∩V )=0.FromthisitfollowsthatPT (V ∩V )=0
α β m+2 α β m+2 α β
for any arrangementof the parentheses T and the variety V ∩V is nilpotent. (cid:3)
α β
We can now prove the main result of this note.
Theorem 2. Over a field of characteristic zero there are countable many almost nilpo-
tent metabelian varieties of at most linear growth and uncountable many almost nilpotent
metabelian varieties of at most quadratic growth.
Proof. Recall that by [11, Theorem 1] every non-nilpotent variety has an almost nilpotent
subvariety. Hence for any real number α, 0 < α < 1, the variety V contains an almost
α
nilpotent subvariety. Let U be such subvariety. Since c (U )≤c (V ), then c (U )≤Cn
α n α n α n α
or c (U ) ≤ Cn2, according as α is rational or irrational, respectively. Hence U has at
n α α
most quadratic growth.
Now, by Proposition 2 for any 0 < α < β < 1 U 6= U , and this says that there
α β
are countable many almost nilpotent metabelian varieties of at most linear growth and
uncountable many almost nilpotent metabelian varieties of at most quadratic growth. (cid:3)
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AN UNCOUNTABLE FAMILY OF ALMOST NILPOTENT VARIETIES OF POLYNOMIAL GROWTH 7
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Departmentof Applied Mathematics,UlyanovskStateUniversity,Ulyanovsk432970,Russia
E-mail address: [email protected]
Dipartimento di Energia, ingegneria dell’Informazione e Modelli Matematici, Universita` di
Palermo, 90128Palermo, Italy
E-mail address: [email protected]