On the growth of the identities of algebras
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irk-123456789-1573812019-06-21T01:27:18Z On the growth of the identities of algebras Giambruno, A. Mishchenko, S. Zaicev, M. 2006 Article On the growth of the identities of algebras / A. Giambruno, S. Mishchenko, M. Zaicev // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 2. — С. 50–60. — Бібліогр.: 24 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/157381 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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On the growth of the identities of algebras / A. Giambruno, S. Mishchenko, M. Zaicev // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 2. — С. 50–60. — Бібліогр.: 24 назв. — англ. |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2006). pp. 50 – 60
c© Journal “Algebra and Discrete Mathematics”
On the growth of the identities of algebras
A. Giambruno, S. Mishchenko, M. Zaicev
Communicated by V. V. Kirichenko
1. Introduction and Preliminaries
Let F be a field of characteristic zero and let A be an algebra over F . One
can associate to the polynomial identities satisfied by the algebra A a nu-
merical sequence cn(A), n = 1, 2, . . . , called the sequence of codimensions
of A ([19]). More precisely, if F{X} is the free algebra on a countable
set X = {x1, x2, . . . } and Pn is the space of multilinear polynomials in
the first n variables, cn(A) is the dimension of Pn modulo the polynomial
identities satisfied by A. The sequence cn(A) gives in some way a measure
of the polynomial relations vanishing in the algebra A and in general, for
non-associative algebras, has overexponential growth. Concerning free
algebras for instance, if F{X} is the free (non-associative) algebra on X,
cn(F{X}) = pnn! where pn = 1
n
(
2n−2
n−1
)
is the n-th Catalan number. For
the free associative algebra F 〈X〉 and the free Lie algebra L〈X〉 we have
cn(F 〈X〉) = n! and cn(L〈X〉) = (n − 1)!.
Since char F = 0, by the well known multilinearization pro-
cess, every T-ideal is determined by its multilinear polynomials.
Hence the T-ideal Id(A) is completely determined by the sequence
of spaces {Pn ∩ Id(A)}n≥1. Now, the symmetric group Sn acts in
a natural way on the space Pn: for σ ∈ Sn, f(x1, . . . , xn) ∈ Pn,
σf(x1, . . . , xn) = f(xσ(1), . . . , xσ(n)). Since for any algebra A, the
subspace Pn ∩ Id(A) is Sn-invariant, this in turn induces a structure of
Sn-module on the space Pn(A) = Pn
Pn∩Id(A) . The Sn-character of Pn(A),
denoted χn(A), is called the n-th cocharacter of the algebra A and
cn(A) = χn(A)(1) = dimF Pn(A)
is the n-th codimension of A.
A. Giambruno, S. Mishchenko, M. Zaicev 51
Since char F = 0, by complete reducibility we can write χn(A) as a
sum of irreducible Sn-characters. Now, it is well known that there is a one-
to-one correspondence between irreducible Sn-characters and partitions
on n (or Young diagrams). Recall that λ = (λ1, . . . , λr) is a partition
of n, and we write λ ⊢ n, if λ1 ≥ · · · ≥ λr > 0 are integers such that
∑r
i=1 λi = n. One usually identifies a partition λ with the corresponding
Young diagram Dλ whose i-th row has length λi. For instance if λ =
(5, 3, 2) ⊢ 10, then the corresponding Young diagram is
There is a well-known and useful formula, called the hook formula, for
computing the degree of an irreducible Sn-character (see for instance
[10]): for λ ⊢ n a partition of n, let χλ denote the corresponding Sn-
character. Then
χλ(1) =
n!
∏
i,j hij
where for any box (i, j) ∈ Dλ, hij = λi + λ
′
j − i − j + 1, with λ
′
=
(λ
′
1, . . . , λ
′
s) the conjugate partition of λ, is the hook number of the cell
(i, j). For instance in the above example h12 = 6, h21 = 4.
We define the hook H(s, t) =
⋃
n≥1{λ = (λ1, λ2, . . .) ⊢ n | λs+1 ≤ t}
where the integer s is called the hand and t the foot of the hook. If A is
an F -algebra, by complete reducibility its n-th cocharacter decomposes
as
χn(A) =
∑
λ⊢n
mλχλ (1)
where mλ ≥ 0 is the multiplicity of χλ in χn(A). Then we shall write
χn(A) ⊆ H(d, l) if λ ∈ H(d, l) for all partitions λ such that mλ 6= 0.
The most important feature of an associative algebra A satisfying a
polynomial identity (PI-algebra) is that cn(A) is exponentially bounded
([19]). There is also a wide class of algebras with exponentially bounded
codimension growth: for instance, if A is any finite dimensional algebra
and dimA = d < ∞, then cn(A) ≤ dn ([1]). Also, any infinite dimen-
sional simple Lie algebra of Cartan type or any affine Kac-Moody algebra
has exponentially bounded codimension growth ([12], [21]).
An interesting remark is that from the hook formula it is easy to
show that if the cocharacter of an algebra A lies in a hook H(s, t) then
the algebra will have exponentially bounded codimension growth. In
case of associative PI-algebras it has been shown that the corresponding
cocharacter lies in some hook H(s, t) ([19]). But this is not a general
52 On the growth of the identities of algebras
phenomenon. For instance if W1 is the infinite-dimensional simple alge-
bra of Cartan type (Witt algebra or Virosoro algebra), then it can be
shown that cn(W1) has exponential growth ([12]) but the corresponding
cocharacter is not contained in any hook ([14]).
In case the sequence of codimensions is exponentially bounded, say
cn(A) ≤ dn, one can construct the bounded sequence n
√
cn(A), n =
1, 2, . . . , and it is an open problem if, in this case, lim
n→∞
n
√
cn(A) ex-
ists. Let us define the PI-exponent of the algebra A as exp(A) =
limn→∞
n
√
cn(A) in case such limit exists.
Let us recall that the sequence of codimensions cn(A) of an algebra
A has
- polynomial growth if there exist constants C and k such that cn(A) ≤
Cnk;
- overpolynomial growth if for any constants C and k there exists n
such that cn(A) > Cnk;
- overexponential growth if for any constants C and b there exists n
such that cn(A) > Cbn.
If the growth is more than polynomial and less than exponential one
says that the sequence cn(A) has intermediate growth.
In the 80’s Amitsur conjectured that for any associative PI-algebra
A, lim
n→∞
n
√
cn(A) exists and is a nonnegative integer. This conjecture was
recently confirmed in [7] and [8].
When A is a Lie algebra, the sequence of codimensions has a much
more involved behavior. Volichenko in [20] showed that a Lie algebra can
have overexponential growth of the codimensions. Starting from this,
Petrogradsky in [18] exhibited a whole scale of overexponential functions
providing the exponential behavior of the identities of polynilpotent Lie
algebras.
Motivated by the good behavior of cn(A) for associative algebras, one
can ask the following question: if a Lie algebra A is such that its se-
quence of codimensions is exponentially bounded, can we infer that the
exponential growth of cn(A) is integer? In [24] it was shown that the
exponential growth of cn(A) is integer for any finite dimensional Lie al-
gebra A. The exponential growth of cn(A) is also integer for any infinite
dimensional Lie algebra A with a nilpotent commutator subalgebra [17].
Anyway the question was answered in the negative by Mishchenko and
Zaicev in [23] by constructing an example of a Lie algebra with exponen-
tial growth strictly between 3 and 4. On the other hand no exponential
growth between 1 and 2 is allowed ([16]).
In this paper we want to review the results proved in the last years
concerning the growth of the codimensions in the general case of non-
A. Giambruno, S. Mishchenko, M. Zaicev 53
associative algebras over a field of characteristic zero.
We remark that in the examples constructed below, all algebras are
left nilpotent of index 2 i.e., they satisfy the identity x1(x2x3) ≡ 0.
In the first part we show that for general algebras whose sequence
of codimensions is exponentially bounded, any real number > 1 can ac-
tually appear. In fact we construct, for any real number α > 1, an
algebra Aα whose sequence of codimensions grows exponentially and
lim
n→∞
n
√
cn(Aα) = α. The details of the proof of this result and some
more results can be found in the original paper [4].
The second part of the paper concerns intermediate growth. For as-
sociative algebras it is known from long time ([11]) that the sequence of
codimensions cannot have intermediate growth. A similar result for Lie
algebras was proved in [15]. For associative superalgebras, associative
algebras with involution and Lie superalgebras it has been shown that
the corresponding codimensions cannot have intermediate growth ([3],
[2], [22]).
Here we show that in the general case of non-associative algebras,
there exist examples of algebras with intermediate growth of the codi-
mensions. To this end for any real number 0 < β < 1 we construct an
algebra whose sequence of codimensions grows as nnβ
. The details of the
proof of this result and some other results can be found in [5]. Finally
two more results are presented: for any finite dimensional algebra the
sequence of codimensions cannot have intermediate growth, and for any
two-dimensional algebra either the exponent is equal to 2 or the growth
of the codimensions is polynomially bounded by n + 1 ([6]).
2. Sturmian or periodic words and real PI-exponent
Given any sequence of integers K = {ki}i≥1 such that ki ≥ 2 for all i, we
can define a (non-associative) algebra A(K) in the following way. We let
A(K) be the algebra over F with basis
{a, b} ∪ Z1 ∪ Z2 ∪ . . .
where
Zi = {z(i)
j | 1 ≤ j ≤ ki}, i = 1, 2, . . . .
and with multiplication table given by
z
(i)
2 a = z
(i)
3 , . . . , z
(i)
ki−1a = z
(i)
ki
, z
(i)
ki
a = z
(i)
1 , i = 1, 2 . . . ,
z
(i)
1 b = z
(i+1)
2 , i = 1, 2, . . .
54 On the growth of the identities of algebras
and all the remaining products are zero.
Some special types of sequences defined below will be of interest for
our purpose. Take w = w1w2 . . . an infinite (associative) word in the
alphabet {0, 1}. Given an integer m ≥ 2, define Km,w = {ki}i≥1 to be
the sequence
ki =
{
m, if wi = 0
m + 1, if wi = 1
and write A(m, w) = A(Km,w).
We recall some of the basic definitions concerning infinite words and
their complexity. In general, given an infinite word w in a finite alphabet,
the complexity Compw of w is the function Compw : N → N, where
Compw(n) is the number of distinct subwords of w of length n.
Recall that an infinite word w = w1w2 · · · in the alphabet {0, 1} is
periodic with period T if wi = wi+T for i = 1, 2, . . . . It is easy to see
that for any such word Compw(n) ≤ T . Moreover, an infinite word w is
called a Sturmian word if Compw(n) = n + 1 for all n ≥ 1 (see [13]).
For a finite word x, the height h(x) of x is the number of letters 1
appearing in x. Also, if |x| denotes the length of the word x, the slope of
x is defined as π(x) = h(x)
|x| . In some cases this definition can be extended
to infinite words in the following way. Let w be some infinite word and
let w(1, n) denote its prefix subword of length n. If the sequence h(w(1,n))
n
converges for n → ∞ and the limit
π(w) = lim
n→∞
h(w(1, n))
n
exists then π(w) it is called the slope of w. It is easy to give examples of
infinite words for which the slope is not defined. Nevertheless for periodic
words and Sturmian words the slope is well defined. The basic properties
of these words are given in the next proposition.
Proposition 1. ([13, 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) for any non-empty subword u of w,
|π(u) − π(w)| ≤ C
|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.
A. Giambruno, S. Mishchenko, M. Zaicev 55
In case w is Sturmian we can take C = 1, and if w is periodic of period
T , then π(w) = h(w(1,T ))
T
.
Recall that, given elements y1, y2, . . . , yn of a non-associative al-
gebra, their left-normed product is defined inductively as y1 · · · yn =
(y1 · · · yn−1)yn. From the definition of the algebra A(K) it easily follows
that only left-normed products of the basis elements of A(K) may be
non-zero. Moreover the only non-zero products are of the type z
(i)
j f(a, b)
for some left-normed monomial f(a, b).
Some conclusions can be easily drawn about the cocharacter sequence
of A(K). For a partition λ = (λ1, . . . , λt) of n let λ′ = (λ′
1, . . . , λ
′
r) de-
note the conjugate partition of λ. Hence h(λ) = λ′
1 is the height of
the Young diagram corresponding to λ. Let Tλ be a λ-tableau and let
eTλ
be the corresponding essential idempotent of FSn. For any polyno-
mial f(x1, . . . , xn) ∈ Pn, the element eTλ
f(x1, . . . , xn) is a linear com-
bination of polynomials each alternating on disjoint sets of variables
of order λ′
1, . . . , λ
′
r, respectively. Since span{z(i)
j } is a two-sided ideal
of A(K) with trivial multiplication of codimension 2, it follows that
eTλ
f(x1, . . . , xn) ≡ 0 in A(K) as soon as h(λ) > 3 or λ3 > 1. This
says that if χn(A(K)) =
∑
λ⊢n mλχλ is the n-th cocharacter of A(K),
then mλ = 0 as soon as h(λ) > 3 or λ3 > 1. In other words
χn(A(K)) = m(n)χ(n) +
∑
λ=(λ1,λ2)⊢n
mλχλ +
∑
λ=(λ1,λ2,1)⊢n
mλχλ.
When studying the cocharacter of the algebra A(K), a special func-
tion Φ : R → R comes into play. This function is defined by
Φ(x) =
1
xx(1 − x)1−x
.
Notice that Φ is continuous in the interval (0, 1
2 ], and Φ(a) < Φ(b) when-
ever a < b. Moreover lim
x→0
Φ(x) = 1 and Φ(1
2) = 2.
At this stage one needs to study the behavior of the n-th cocharacter
of A(m, w). Roughly speaking one proves that all characters χλ whose
diagram λ has long second row, do not participate in χn(A(m, w)). This
fact is exploited in the next lemmas in order to get an upper bound and
a lower bound for cn(A(m, w)).
Lemma 1. Let w be a Sturmian or periodic word with slope π(w) = α,
let A = A(m, w) and let β = 1
m+α
. Then, given any ε > 0 there exists
N = N(ε) such that for all n ≥ N , the (n + 1)-th codimension of A
satisfies
cn+1(A) ≥ 1
2m+1
√
πn3
Φ(β + ε)n.
56 On the growth of the identities of algebras
Lemma 2. Let w be a Sturmian or periodic word with slope α and
let A = A(m, w). If β = 1
m+α
, then cn+1(A) ≤ 3(m + 1)(n + 2)5Φ(β)n.
Putting together Lemma 1 and Lemma 2 it is clear that for the alge-
bras A(m, w) the PI-exponent exists and equals Φ(β). We record this in
the following.
Proposition 2. Let w be an infinite Sturmian or periodic word with
slope α, 0 < α < 1. If m ≥ 2 then for the algebra A = A(m, w) the
PI-exponent exists and exp(A) = Φ(β) where β = 1
m+α
.
Recalling that the function Φ is continuous and Φ((0, 1
2)) = (1, 2), we
immediately obtain.
Corollary 1. For any real number d, 1 < d < 2, there exists an
algebra A such that exp(A) = d.
All algebras A(m, w) constructed above are infinite dimensional. In
case the word w is periodic we can actually construct a finite dimensional
algebra B such that Id(B) = Id(A(m, w)) (and exp(B) = exp(A(m, w))).
The construction is the following. Recall that given an infinite word on
{0, 1} and m ≥ 2, the sequence Km,w = {ki}i≥1 is defined by ki =
{
m, if wi = 0
m + 1, if wi = 1
. Then, if w is an infinite periodic word of period
T , we define B(K) as the algebra over F with basis
{a, b} ∪ Z1 ∪ Z2 ∪ · · · ∪ ZT
where
Zi = {z(i)
j | 1 ≤ j ≤ ki}, i = 1, 2, . . . , T,
and multiplication table given by
z
(i)
2 a = z
(i)
3 , . . . , z
(i)
ki−1a = z
(i)
ki
, z
(i)
ki
a = z
(i)
1 , i = 1, 2 . . . ,
z
(i)
1 b = z
(i+1)
2 , i = 1, 2, . . . , (T − 1)
and
z
(T )
1 b = z
(1)
2 .
All the remaining products are zero.
The following results are obvious consequences of the definition of the
algebra B(K).
Proposition 3. The algebras A(K) and B(K) satisfy the same iden-
tities.
From this result we have
A. Giambruno, S. Mishchenko, M. Zaicev 57
Proposition 4. For any rational number β, 0 < β ≤ 1
2 , there exists
a finite dimensional algebra B such that exp(B) = Φ(β).
We next wish to extend Proposition 2 and Corollary 1 to all real
numbers > 1 i.e., we want to construct, for any real number α > 1 an
algebra A such that exp(A) = α. We can accomplish this by constructing
an appropriate algebra B and then by gluing, in an appropriate way, B
to one of the algebras A(m, w) constructed above.
Given any positive integer d we define a non-associative algebra B =
B(d) as follows: B has basis {u1, . . . ud, s1, . . . , sd} with multiplication
table given by
s1u1 = u2, . . . sd−1ud−1 = ud, sdud = u1,
and all other products are zero.
Starting with A(K) and B we next define an algebra A(K, d) which
will contain both A(K) and B as subalgebras.
Let W be the vector space spanned by the set {wi,j | 1 ≤ i ≤ d, j ≥ 1}
and let A(K, d) be the algebra which is the vector space direct sum of
A(K), B and W ,
A(K, d) = A(K) ⊕ B ⊕ W.
The multiplication in A(K, d) is induced by the multiplication in A(K), B
and usz
i
j = wsi, 1 ≤ s ≤ d, 1 ≤ j ≤ ki, i ≥ 1, and all other products are
zero.
We start by studying the identities of B = B(d).
Lemma 3. The algebra B satisfies the right-normed identity
y1(x1 · · · (xd−1(y2xd)) . . . ) ≡ y2(x1 · · · (xd−1(y1xd)) . . . )
and the left-normed identity x1x2x3 ≡ 0.
Proposition 5. Let m ≥ 2 and let w be a periodic or Sturmian
word. Then the PI-exponent of the algebra A(Km,w, d) exists and
exp(A(Km,w, d)) = d + δ where δ = exp(A(Km,w)).
At last we formulate the main results about the PI-exponent ([4]).
Theorem 1. For any real number t ≥ 1 there exists an algebra A
such that
exp(A) = t.
Theorem 2. For any 1 ≤ α < β there exists a finite dimensional
algebra A such that
α < exp(A) < β.
58 On the growth of the identities of algebras
3. Constructing intermediate growth
As we mentioned in the introduction, no intermediate growth is allowed
for the codimensions of an associative algebra or a Lie algebra. Neverthe-
less this is not a general phenomenon. In fact we shall next construct non-
associative algebras whose codimensions have intermediate growth. Re-
call that given an algebra A with n-th cocharacter χn(A) =
∑
λ⊢n mλχλ,
then ln(A) =
∑
λ⊢n mλ is the n-th colength of A.
Given a sequence K = {ki}i≥1 we can associate to K a real valued
function ρ such that ki = ρ(i), i = 1, 2, . . . . In this case we also write
A(K) = A(ρ).
Let now ρ be a polynomially bounded real function i.e.,
ρ(x) ≤ cxβ
for all positive real numbers x, where β > 0 and c > 0 are constants. We
next find some condition on such function ρ, so that the algebra A(ρ) has
polynomially bounded colength sequence.
Lemma 4. Let ρ be a polynomially bounded monotone function,
ρ(x) ≤ cxβ, for all x ∈ R
+, and β > 0, c > 0 constants. If limx→∞ ρ(x) =
∞ then, for n large enough, the colength sequence of the algebra A(ρ) sat-
isfies
ln(A(ρ)) ≤ (n + 1)3
(
3n + 3c
(
2n
c
)
β+1
β
)
.
We now specialize the polynomially bounded function ρ to a function
that behaves like γxγ−1 for some fixed γ > 1.
Let β be a real number such that 0 < β < 1 and let γ = 1
β
. Then we
define A = A(β) as the algebra A(K) where the sequence K = {ki}i≥1
is defined by the relation k1 + · · · + kt = [tγ ], for all t ≥ 1, where [x] is
the integer part of x. We can show that such algebras have intermediate
growth of the codimensions. In fact we have ([5])
Theorem 3. For any real number β with 0 < β < 1, let A = A(K)
where K = {ki}i≥1 is defined by the relation k1 + · · · + kt = [t
1
β ], for all
t ≥ 1. Then the sequence of codimensions of A satisfies
lim
n→∞
logn logn cn(A) = β.
Hence, cn(A) asymptotically equals nnβ
.
The above algebras are infinite dimensional. Hence it is natural to ask
if one can construct finite dimensional algebras with such property. The
A. Giambruno, S. Mishchenko, M. Zaicev 59
answer is negative since, as we shall see, any finite dimensional algebra
cannot have intermediate growth of the codimensions.
Concerning the colength sequence of an arbitrary finite dimensional
algebra it can be shown that is polynomially bounded and depends only
on dimA. In fact the following holds.
Theorem 4. Let A be a finite dimensional algebra, dimA = d. Then
ln(A) ≤ d(n + 1)d2+d.
In the next theorem we show that no intermediate growth is allowed
for a finite dimensional algebra A and we can exhibit a lower bound for
the exponential growth depending only on dimA ([5]).
Theorem 5. Let A be a finite dimensional algebra of overpolynomial
codimension growth and let dimA = d. Then cn(A) > 1
n2 2
n
3d3 , for all n
large enough.
More precise information can be obtained in case of an arbitrary two-
dimensional algebra. In fact we have the following (see [6]).
Theorem 6. Let A be a two-dimensional algebra over an algebraically
closed field of characteristic zero. Then cn(A) ≤ n + 1 or exp(A) = 2.
References
[1] Y. Bahturin and V. Drensky, Graded polynomial identities of matrices, Linear
Algebra Appl. 357 (2002), 15-34.
[2] A. Giambruno and S. Mishchenko, On star-varieties with almost polynomial
growth Algebra Colloq. 8 (2001), no. 1, 33–42.
[3] A. Giambruno, S. Mishchenko and M. Zaicev, Polynomial identities on superal-
gebras and almost polynomial growth. Special issue dedicated to Alexei Ivanovich
Kostrikin Comm. Algebra 29 (2001), no. 9, 3787–3800.
[4] A. Giambruno, S. Mishchenko and M. Zaicev, Codimensions of Algebras and
Growth Functions, Preprint N. 264 Dipt. Mat. Appl. Università di Palermo (2004),
1-19.
[5] A. Giambruno, S. Mishchenko and M. Zaicev, Algebras with intermediate growth
of the codimensions Adv. Applied Math. (to appear).
[6] A. Giambruno, S. Mishchenko and M. Zaicev, Codimension growth of two dimen-
sional algebras (preprint).
[7] A. Giambruno and M. Zaicev, On codimension growth of finitely generated asso-
ciative algebras, Adv. Math. 140 (1998), 145-155.
[8] A. Giambruno and M. Zaicev, Exponential codimension growth of P.I. algebras:
an exact estimate Adv. Math. 142 (1999), 221-243.
[9] A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods,
AMS, Mathematical Surveys and Monographs, Vol. 122, Providence R.I., 2005.
60 On the growth of the identities of algebras
[10] G. James and A. Kerber, The representation theory of the symmetric group, En-
cyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley, Lon-
don, 1981.
[11] A. R. Kemer, T-ideals with power growth of the codimensions are Specht (Rus-
sian), Sibirskii Matematicheskii Zhurnal 19 (1978), 54-69; translation in Siberian
Math. J. 19 (1978), 37-48.
[12] S. P. Mishchenko, On the problem of the Engel property, (Russian) Mat. Sb. 124
(166) (1984), 56–67; translation in Math. USSR Sb. 52 (1985).
[13] M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics
and Its Applications vol 90, Cambridge University Press, Cambridge, 2002.
[14] S. P. Mishchenko, Growth in varieties of Lie algebras, English. Russian original
Russ. Math. Surv. 45, No.6, 27-52 (1990); translation from Usp. Mat. Nauk 45,
No.6(276), 25-45 (1990).
[15] S. P. Mishchenko, On varieties of Lie algebras of intermediate growth (Russian)
Vests̄ıAkad. Navuk BSSR Ser. F̄ız.-Mat. Navuk 126 1987, no. 2, 42–45.
[16] S. P. Mishchenko, Lower bounds on the dimensions of irreducible representations
of symmetric groups and of the exponents of the exponential of varieties of Lie
algebras (Russian) Mat. Sb. 187 (1996), 83–94; translation in Sb. Math. 187
(1996), 81–92.
[17] S. P. Mishchenko and V. M. Petrogradsky, Exponents of varieties of Lie algebras
with a nilpotent commutator subalgebra Communications in Algebra. 1999. V.27.
N 5. P.2223–2230.
[18] V. M. Petrogradsky, Growth of polynilpotent varieties of Lie algebras, and rapidly
increasing entire functions (Russian) Mat. Sb. 188 (1997), no. 6, 119–138; trans-
lation in Sb. Math. 188 (1997), no. 6, 913–931
[19] A. Regev, Existence of identities in A ⊗ B, Israel J. Math. 11 (1972), 131-152.
[20] I. B. Volichenko, Varieties of Lie algebras with the identity
[[X1, X2, X3], [X4, X5, X6]] = 0 over a field of characteristic zero (Russian)
Sibirsk. Mat. Zh. 25 (1984), no. 3, 40–54.
[21] M. V. Zaicev, Varieties and identities of affine Kac-Moody algebras, Methods in
ring theory, 303-314, Lecture Notes in Pure and Appl. Math., Vol. 198, Dekker,
New York, 1998.
[22] M. V. Zaicev and S. P. Mishchenko, A criterion for the polynomial growth of va-
rieties of Lie superalgebras (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998),
no. 5, 103–116; translation in Izv. Math. 62 (1998), no. 5, 953–967.
[23] M. V. Zaicev and S. P. Mishchenko, An example of a variety of Lie algebras with
a fractional exponent Algebra, 11. J. Math. Sci. (New York) 93 (1999), no. 6,
977–982.
[24] M. V. Zaicev, Integrality of exponents of growth of identities of finite-dimensional
Lie algebras (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), 23-48; trans-
lation in Izv. Math. 66 (2002), 463-487.
Contact information
A. Giambruno Università di Palermo, Palermo, Italy
E-Mail: agiambr@unipa.it
A. Giambruno, S. Mishchenko, M. Zaicev 61
S. Mishchenko Ulyanovsk State University, Ulyanovsk,
Russia
E-Mail: mishchenkosp@ulsu.ru
M. Zaicev Moscow State University, Moscow, Russia
E-Mail: zaicev@mech.math.msu.su
Received by the editors: 19.12.2005
and in final form 30.06.2006.
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