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\documentclass{era-l}

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\dateposted{June 6, 2000}
\pagespan{40}{44}
\PII{S 1079-6762(00)00078-0}
\def\copyrightyear{2000}
%\def\copyrightyear{2000}
\copyrightinfo{2000}{American Mathematical Society}

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\begin{document}

\title{Minimal varieties of algebras of exponential growth}
\author{A. Giambruno}
\address{Dipartimento di Matematica ed Applicazioni, Universit\`a di
Palermo, 90123 Palermo, Italy}
\email{a.giambruno@unipa.it}
\thanks{The first author was partially supported by MURST of Italy; the
second author was partially supported by the RFBR grants 99-01-00233 and
96-15-96050.}

\author{M. Zaicev}
\address{Department of Algebra, Faculty of Mathematics and Mechanics,
Moscow State University,  Moscow 119899, Russia}
\email{zaicev@mech.math.msu.su}

\commby{Efim Zelmanov}
\date{October 4, 1999}

\subjclass[2000]{Primary 16R10, 16P90}

\keywords{Varieties of algebras, polynomial identities}

\begin{abstract}
The exponent of a variety of algebras over a field of characteristic
zero has been recently proved to be an integer.
Through this scale we can now classify all minimal varieties of a given
exponent and of finite basic rank.
As a consequence we describe the corresponding T-ideals of the  free
algebra, and we compute the asymptotics of the related codimension
sequences.
We then verify in this setting  some known conjectures.
\end{abstract}

\maketitle

\section{Introduction}
In this note we deal with the classification of varieties of associative
algebras over a field of characteristic zero.
A natural scale is provided by the so-called {\it exponent} of a variety,
which has been  recently proved to be an integer.

We recall the terminology and the basic definitions.
Throughout,  $F$ will be a field of characteristic zero, $X=\{x_1,x_2,
\dots \}$ a countable set and $F\langle X\rangle$ the free algebra on
$X$ over $F.$
Recall that  a  polynomial $f(x_1,\dots, x_n) \in F\langle X\rangle$ is
an identity for an  associative $F$-algebra $A$  if  $f(a_1, \dots,
a_n)=0$  for all $a_1, \dots, a_n\in A.$
The set of all polynomial identities of  $A$ is a T-ideal $\
\text{Id}(A)$ of $F\langle X\rangle$, i.e., an ideal invariant under all
endomorphisms of $F\langle X\rangle$, and it is easy to see that all
T-ideals of $F\langle X\rangle$ are of this type.

In order to study  the properties shared by all algebras with a given
set of polynomial identities or the T-ideals  of $F\langle X\rangle$,
the language of varieties seems the most appropriate.
Recall that the class $\mathcal{V}$ of all  $F$-algebras satisfying all
the polynomial identities of a T-ideal $I$ is called the variety
associated to $I.$
We write $I=\text{Id}(\mathcal{V})$ and, in case $\mathcal{V}$ is
generated by   an algebra $A$, we write $\mathcal{V} = \text{var}(A)$.
It is clear that  $F\langle X\rangle / \text{Id}(\mathcal{V})$ belongs
to the variety $\mathcal{V}$ and is the relatively free algebra of
countable rank of $\mathcal{V}$.

An important invariant of a variety $\mathcal{V}$  is the so-called
growth function  or codimension growth, which  we now describe: for every
$n\ge 1$ let $P_n$ be the space of all multilinear polynomials in the
variables $x_1,\dots, x_n.$
Since $\text{char }F=0$, by the well-known multilinearization or
polarization process, a T-ideal $\ \text{Id}(\mathcal{V})$ is determined
by the multilinear polynomials it contains; hence the relatively free
algebra $F\langle X\rangle / \text{Id}(\mathcal{V})$ is determined by
the sequence of subspaces $\{P_n/(P_n \cap
\text{Id}(\mathcal{V}))\}_{n\ge 1}$.
  The integer $c_n(\mathcal{V}) = \dim P_n/(P_n \cap
\text{Id}(\mathcal{V}))$ is called the $n$th codimension  of
$\mathcal{V}$, and the growth function determined by the integral
sequence $\{
c_n(\mathcal{V})\}_{n\ge 1}$ is the growth of the variety $\mathcal{V}.$

It is well known that if $\mathcal{V}$  is a nontrivial variety, then
the sequence of codimensions is exponentially bounded (\cite{reg}),
i.e., there exist constants $\alpha, a >0$ such that $c_n(\mathcal{V})
\le \alpha a^n$ for all $n$.
 Kemer in \cite{kem1} characterized the varieties with a polynomially
bounded codimension sequence (see also \cite{giazai3}).
From his description it follows that there exists no variety with
intermediate growth of the codimensions between polynomial and
exponential.

 To capture the exponential behavior  of this sequence one defines
$\overline{\text{Exp}(\mathcal{V})} = \lim\sup_{n\rightarrow
\infty}\root n \of{c_n(\mathcal{V})}, \quad
\underline{\text{Exp}(\mathcal{V})} = \lim\inf_{n\rightarrow
\infty}\root n \of{c_n(\mathcal{V})}$ and, in the case of equality,
$\text{Exp}(\mathcal{V})= \overline{\text{Exp}(\mathcal{V})}=
\underline{\text{Exp}(\mathcal{V})}.$
In the $80$'s it was conjectured by Amitsur that for any nonnilpotent
proper variety $\mathcal{V}, \ \text{Exp}(\mathcal{V})$ exists and is an
integer.
This conjecture  has been recently verified.
 For a nilpotent variety $\mathcal{V},$ define
$\text{Exp}(\mathcal{V})=0$.
Then we have

\begin{thm}[\cite{giazai1}, \cite{giazai2}]
For any proper variety $\mathcal{V}, \  \text{Exp}(\mathcal{V})$ exists
and is an integer.
\end{thm}


The integer $\text{Exp}(\mathcal{V})$ is called the exponent of the
variety $\mathcal{V}.$

Among varieties of a given exponent a prominent role is played by the 
so-called minimal varieties.
A variety of exponent $d$ is minimal if every proper subvariety has
exponent strictly less than $d.$
From  Kemer's result \cite{kem1} it turns out that  the infinite
dimensional Grassmann algebra $G$ and the algebra of $2\times 2$ upper
triangular matrices $UT_2$  generate the only two minimal varieties of
exponent $2.$
Moreover, Drensky in \cite{dren1} and \cite{dren2} has proved that any
polynomial of the type $[x_1,x_2]\cdots [x_{2n-1}, x_{2n}]$ or
$[x_1,x_2,x_3]\cdots [x_{3n-2}, x_{3n-1},x_{3n}]$ determines a minimal
variety.

Here we shall  classify all minimal varieties of finite basic rank.
Recall that a variety $\mathcal{V}$ has finite basic rank if $\
\mathcal{V}= \text{var}(A)$, where $A$ is a finitely generated algebra.
According to a result of Kemer   such a variety can be generated by a
finite dimensional algebra; another equivalent formulation is that
$\mathcal{V}$ satisfies all Capelli polynomials of some rank (see
\cite{kem2}).

We shall describe the minimal varieties of finite basic rank  in terms
of generating algebras and in terms of identities.
As a consequence we shall verify for these varieties a conjecture of
Drensky stating that a variety $\mathcal{V}$ is minimal if and only if
$\text{Id}(\mathcal{V})$  is a product of so-called verbally prime
T-ideals.

As a corollary we shall also find the exact asymptotics for the
codimension sequence of such minimal varieties.
Recall that two functions of  a natural variable $f(n)$ and $g(n)$ are
asymptotically equal and we write  $f(n) \simeq g(n)$, if $\lim_{n \to
\infty} \frac{f(n)}{g(n)} =1.$

Regev and Berele (\cite{reg2}, \cite{br1})  computed the asymptotics for
some important classes of algebras, in particular for the matrix
algebras over $F$.
In all cases they found that
\begin{equation}
c_n(A)\simeq\alpha n^gd^n,
\end{equation}
where $d\ge 1$ is an integer, $g$ is a half-integer and $\alpha$ is a
constant.
Generalizing Amitsur's conjecture Regev formulated the hypothesis that a
relation of type (1.1) holds for any algebra $A$ satisfying a
nontrivial identity (PI-algebra).
Our description of minimal varieties together with results of Berele,
Regev and Lewin allows us to explicitly compute the asymptotics  of
$c_n(A)$ for any upper block-triangular matrix algebra over $F$.
This confirms Regev's conjecture for any minimal variety of finite basic
rank.

  Finally  we discuss the relationship between growth of algebras and
growth of identities.
We find some relations between the exponent of the codimension growth
and the Gelfand-Kirillov dimension of a finitely generated relatively
free algebra.



\section{Upper block-triangular matrix algebras}

Let $d_1,\dots,d_m$ be positive integers and let $UT(d_1,\dots,d_m)$
denote the algebra of all upper block-triangular matrices over $F$ of
the type
\begin{equation*}
\begin{pmatrix}
M_{d_1}(F) &  &  & * \\ 0 & \ddots & &  \\  \vdots &   &  & \\  0 &
\cdots  & 0 & M_{d_m}(F) \\
\end{pmatrix},
\end{equation*}
where $M_{d_i}(F)$ is the algebra of $d_i \times d_i$ matrices over $F$.

The Wedderburn-Malcev decomposition of this algebra is readily written
as
\begin{equation*}
UT(d_1,\dots,d_m) \cong M_{d_1}(F) \oplus \cdots \oplus M_{d_m}(F) + J,
\end{equation*}
where $J$ is the Jacobson radical of $UT(d_1,\dots,d_m)$ and it consists
of  all strictly upper block-triangular matrices.
Invoking \cite{giazai1} it is easy to see that
\begin{equation*}\text{Exp}(UT(d_1,\dots,d_m))=d_1^2+\cdots +d_m^2\;.\end{equation*}

The next result shows that algebras of upper block-triangular matrices arise
naturally in any finite dimensional algebra.


\begin{lemma}
Let $F$ be an algebraically closed field of characteristic zero.
If $A$ is a finite dimensional algebra over $F$ and $\text{Exp} (A)=d\ge
2$, then there exists a subalgebra of $A$ isomorphic to
$UT(d_1,\dots,d_m)$ with $d_1^2+\cdots +d_m^2=d$.
\end{lemma}


What about comparing the identities of two algebras of upper
block-triangular matrices?
Let $A=UT(d_1,\dots,d_m)$ and $B=UT(q_1,\dots,q_s)$ be two such
algebras and suppose that $q=q_1+\cdots +q_s\le d=d_1+\cdots +d_m$.
Then one can consider the canonical embeddings of $A$ and $B$ into
$M_d(F)$.
In this case $B$ lies inside $A$ if and only if for any
$j\in\{1,\dots,m\}$ there exists $i\in\{1,\dots,s\}$ such that
$q_1+\cdots +q_i=d_1+\cdots +d_j$ provided that $d_1+\cdots +d_j\le q$.
It turns out that $B$ satisfies all identities of $A$ if and only if $B$
is a subalgebra of $A$.


\begin{lemma}
Let $A=UT(d_1,\dots,d_m)$ and $B=UT(q_1,\dots,q_s)$ be two upper
block-triangular matrix algebras over $F$.
Then $B\in \text{var}(A)$ if and only if $B\subseteq A$.
\end{lemma}


It is possible to compute the T-ideal of identities of an upper
block-triangular matrix algebra.
The main tool for this proof is a result of Lewin \cite{lew}; it turns
out that such a T-ideal is a product of T-ideals of identities of matrix
algebras.

\begin{theorem} \label{id}
The T-ideal of identities of the algebra of upper block-triangular
matrices $UT(d_1,\dots,d_m)$  is
\begin{equation*}
\text{Id}(UT(d_1,\dots,d_m)) = \text{Id}(M_{d_1}(F)) \cdots
\text{Id}(M_{d_m}(F)).
\end{equation*}
\end{theorem}





\section{Minimal varieties of finite basic rank}


The algebras of upper block-triangular matrices provide generators for
the minimal varieties of finite basic rank.
In fact we can prove

\begin{theorem} \label{th}
Let $\mathcal{V}$ be a variety of finite basic rank and suppose
$\text{Exp} (\mathcal{V})=d\ge 2$.
Then $\mathcal{V}$ is a minimal variety of exponent $d$ if and only if
$\mathcal{V}$ is generated by some upper block-triangular matrix algebra
$UT(d_1,\dots,d_m)$ with $d_1^2+\cdots +d_m^2=d$.
\end{theorem}

Theorem \ref{th} implies the following interesting remarks:

\begin{corollary}
For any positive integer $k$ there exists a variety $\mathcal{V}$ such
that $\ \text{Exp} (\mathcal{V})=d$ but $\ \text{Exp} (\mathcal{U}) \le
d-k$ for any proper subvariety $\mathcal{U}\subset\mathcal{V}$.
\end{corollary}


\begin{corollary}
For any integer $d\ge 2$, the number of minimal varieties of finite
basic rank and of exponent $d$ is finite.
\end{corollary}


Kemer introduced the notion of verbally prime T-ideal.
Recall that $I$ is such a T-ideal if $fg\in I$ implies that either $f\in
I$ or $g\in I$, where $f$ and $g$ are two polynomials in $F\langle
X\rangle$  in disjoint sets of variables.
It turns out that for a finitely generated algebra $A, \ \text{Id}(A)$
is verbally prime if and only if $\text{Id}(A) = \text{Id}(M_d(F))$ for
some $d\ge 1$.

Drensky in \cite{dren1} conjectured that a variety $\mathcal{V}$ is
minimal if and only if $\ \text{Id}(\mathcal{V})$ is a product of
verbally prime T-ideals.
As a consequence of Theorem \ref{id} we are now able to confirm this
conjecture for varieties of finite basic rank.


\begin{corollary}
Let $\mathcal{V}$ be a variety of finite basic rank and let
$I=\text{Id}(\mathcal{V})$.
Then $\mathcal{V}$ is minimal of exponent $d\ge 2$ if and only if
$I=I_1\cdots I_k$, where $I_1,\dots,I_k$ are verbally prime T-ideals.
\end{corollary}


Another application of Theorem \ref{id}  is a precise asymptotic formula
for the codimension growth of upper block-triangular matrix algebras.
From \cite{reg2} and \cite[Theorem 1.4]{br2} immediately follows

\begin{theorem}
Let $A=UT(d_1,\dots,d_m)$. Then the asymptotic behavior of its
codimensions is described by the formula
\begin{equation*}
c_n(A)\simeq\alpha n^g d^n
\end{equation*}
with $d=d_1^2+\cdots +d_m^2$, $g=-\frac{1}{2}\left(\sum_{i=1}^m d_i^2 -
3m+2 \right)$ and
\begin{equation*}
\alpha = \alpha_1\cdots \alpha_m \frac{d_1^{g_1}\cdots d_m^{g_m}} {(
d_1^2+\cdots + d_m^2)^g},
\end{equation*}
where $g_i=-(d_i^2-1)$,
\begin{equation*}
\alpha_i = \left(\frac{1}{\sqrt{2\pi}}\right)^{d_i-1}\cdot
\left(\frac{1}{2}\right)^{\frac{1}{2}(d_i^2-1)} \cdot1 !\cdot 2!\cdots
(d_i-1)!d_i^{\frac{1}{2}(d_i^2+4)}.
\end{equation*}
\end{theorem}

Hence the previous theorem confirms Regev's conjecture for minimal
varieties of finite basic rank.



The Gelfand-Kirillov dimension of  PI-algebras has been extensively
investigated in the last years (see, for example, \cite{dren3}).
We point out some relations between such dimension of a relatively free
algebra and the codimension growth of the corresponding variety (see
also \cite{pro} and \cite{ber}).
For a finitely generated algebra $A$ we denote by $\text{GKdim }A\ $ the
Gelfand-Kirillov dimension of $A$.


\begin{theorem}
Let $\mathcal{V}$ be a nontrivial variety  and $R_m=R_m(\mathcal{V})$
the relatively free $\mathcal{V}$-algebra with $m$ generators.
Then
\begin{equation*}
\text{GKdim } R_m\ge (m-1)\text{Exp}(R_m)+1.
\end{equation*}
If $\mathcal{V}=\text{var } UT(d_1,\dots,d_k)$ and $m\ge 2$, then
\begin{equation*}
\text{GKdim } R_m = (m-1)\text{Exp}(R_m)+k= (m-1)(d_1^2+\cdots
+d_k^2)+k.
\end{equation*}
\end{theorem}


\bibliographystyle{amsplain}
\begin{thebibliography}{99}


\bibitem{ber} A. Berele, \textit{Generic verbally prime PI-algebras and
their GK-dimension}, Comm. Algebra \textbf{21} (1993), 1478--1504.
\MR{94f:16038}
\bibitem{br1} A. Berele and A. Regev, \textit{On the codimensions of the
verbally prime P.I. algebras}, Israel J. Math. \textbf{91} (1995),
239--247.
\MR{96g:16028}
\bibitem{br2} A. Berele and A. Regev, \textit{Codimensions of products
and intersections of verbally prime T-ideals}, Israel J. Math.
\textbf{103} (1998), 17--28.
\MR{99b:16037}
\bibitem {dren1} V. Drensky, \textit{Extremal varieties of algebras I},
Serdica \textbf{13} (1987), 320--332. (Russian)
\MR{90e:08010}
\bibitem {dren2} V. Drensky, \textit{Extremal varieties of algebras II},
Serdica \textbf{14} (1988), 20--27. (Russian)
\MR{90e:08011}
\bibitem {dren3} V. Drensky, \textit{Gelfand-Kirillov dimension of
PI-algebras}, In: Methods in Ring Theory, Lect. Notes in Pure and
Appl. Math., vol. 198,  Marcel Dekker, New York, 1998, pp. 97--113.
\bibitem {giazai1} A. Giambruno and M. Zaicev \textit{On codimension
growth of finitely generated associative algebras,} Adv. Math.
\textbf{140} (1998), 145--155.
\MR{99k:16049}
 \bibitem{giazai2}  A. Giambruno and M. Zaicev, \textit{Exponential
codimension growth of PI-algebras: an exact estimate,}  Adv. Math.
\textbf{142} (1999), 221--243.
\MR{2000a:16048}
 \bibitem{giazai3}  A. Giambruno and M. Zaicev, \textit{A
characterization of algebras with polynomial growth of the
codimensions},  Proc. Amer. Math. Soc. (to appear).
\bibitem{kem1} A. Kemer, \textit{T-ideals with power growth of the
codimensions are Specht}, Sibirskii Matema\-ti\-che\-skii Zhurnal,
{\bf 19} (1978), 54--69; English  translation: Siberian Math. J.
\textbf{19} (1978), 37--48.
\MR{57:6070}
\bibitem{kem2} A. Kemer, \textit{Ideals of identities of associative
algebras}, AMS Translations of Mathematical Monographs, Vol. 87, 1988.
\MR{92f:16031}
\bibitem{lew} J. Lewin, \textit{A matrix representation for  associative
algebras. I}, Trans. Amer. Math. Soc. \textbf{188} (1974),  293--308.
\MR{49:2848}
\bibitem{pro} C. Procesi, \textit{Non-commutative affine rings},  Atti
Acc.  Naz. Lincei, Ser. VIII \textbf{8} (1967), 239--255.
\MR{37:256}
\bibitem{reg} A. Regev, \textit{Existence of identities in $A \otimes
B$,} Israel J. Math. \textbf{11} (1972), 131--152.
\MR{47:3442}
\bibitem{reg2} A. Regev, \textit{Codimensions and trace codimensions of
matrices are asymptotically equal}, Israel J. Math. \textbf{47} (1984),
246--250.
\MR{85j:16024}
\end{thebibliography}
\end{document}

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