An ensemble-acute lymphoblastic leukemia model for acute lymphoblastic leukemia image classification

1.   Introduction

We consider numerical invariants associated with polynomial identities of algebras over a field of characteristic zero. Given an algebra $A$, one can construct a sequence of non-negative integers $\{c_n(A)\}, n = 1,2,\ldots$, called the codimensions of $A$, which is an important numerical characteristic of identical relations of $A$. In general, the sequence $\{c_n(A)\}$ grows faster than $n!$. However, there is a wide class of algebras with exponentially bounded codimension growth. This class includes all associative PI-algebras [2], all finite-dimensional algebras [2], Kac-Moody algebras [12], infinite-dimensional simple Lie algebras of Cartan type [9], and many others. If the sequence $\{c_n(A)\}$ is exponentially bounded then the following natural question arises: does the limit

exist and what are its possible values? In case of existence, the limit (1.1) is called the PI-exponent of $A$, denoted as $exp(A)$. At the end of 1980's, Amitsur conjectured that for any associative PI-algebra, the limit (1.1) exists and is a non-negative integer. Amitsur's conjecture was confirmed in [5,6]. Later, Amitsur's conjecture was also confirmed for finite-dimensional Lie and Jordan algebras [4,15]. Existence of $exp(A)$ was also proved for all finite-dimensional simple algebras [8] and many others.

Nevertheless, the answer to Amitsur's question in the general case is negative: a counterexample was presented in [14]. Namely, for any real $\alpha>1$, an algebra $R_\alpha$ was constructed such that the lower limit of $\root n \of{c_n(A)}$ is equal to $1$, whereas the upper limit is equal to $\alpha$. It now looks natural to describe classes of algebras in which for any algebra $A$, its PI-exponent $exp(A)$ exists. One of the candidates is the class of all finite-dimensional algebras. Another one is the class of so-called special Lie algebras. The next interesting class consists of unital algebras, it contains in particular, all algebras with an external unit. Given an algebra $A$, we denote by $A^\sharp$ the algebra obtained from $A$ by adjoining the external unit. There is a number of papers where the existence of $exp(A^\sharp)$ has been proved, provided that $exp(A)$ exists [11,16,17]. Moreover, in all these cases, $exp(A^\sharp) = exp(A)+1$.

The main goal of the present paper is to construct a series of unital algebras such that $exp(A)$ does not exist, although the sequence $\{c_n(A)$ is exponentially bounded (see Theorem 3.1 and Corollary 3.1 below). All details about polynomial identities and their numerical characteristics can be found in [1,3,7].

2.   Definitions and preliminary structures

Let $A$ be an algebra over a field $F$ and let $F\{X\}$ be a free $F$-algebra with an infinite set $X$ of free generators. The set $Id(A)\subset F\{X\}$ of all identities of $A$ forms an ideal of $F\{X\}$. Denote by $P_n = P_n(x_1,\ldots,x_n)$ the subspace of $F\{X\}$ of all multilinear polynomials on $x_1,\ldots,x_n\in X$. Then $P_n\cap Id(A)$ is actually the set of all multilinear identities of $A$ of degree $n$. An important numerical characteristic of $Id(A)$ is the sequence of non-negative integers $\{c_n(A)\},n = 1,2,\ldots\; ,$ where

If the sequence $\{c_n(A)\}$ is exponentially bounded, then the lower and the upper PI-exponents of $A$, defined as follows

are well-defined. An existence of ordinary PI-exponent (1.1) is equivalent to the equality $\underline{exp}(A) = \overline{exp}(A)$.

In [14], an algebra $R = R(\alpha)$ such that $\underline{exp}(R) = 1,\overline{exp}(R) = \alpha$, was constructed for any real $\alpha>0$. Slightly modifying the construction from [14], we want to get for any real $\alpha>2$, an algebra $R_\alpha$ with $\underline{exp}(R_\alpha)^\sharp = 2$ and $\alpha\le \overline{exp}(R^\sharp)\le\alpha+1$.

Clearly, polynomial identities of $A^\sharp$ strongly depend on the identities of $A$. In particular, we make the following observation. Note that if $f = f(x_1,\ldots, x_n)$ is a multilinear polynomial from $F\{X\}$ then $f(1+x_1,\ldots,1+x_n)\in F\{X\}^\sharp$ is the sum

where $f_{i_1,\ldots,i_k}$ is a multilinear polynomial on $x_{i_1},\ldots,x_{i_k}$ obtained from $f$ by replacing all $x_j,j\ne i_1,\ldots,i_k$ with $1$.

Remark 2.1. A multilinear polynomial $f = f(x_1,\ldots,x_n)$ is an identity of $A^\sharp$ if and only if all of its components $f_{i_1,\ldots,i_k}$ on the left hand side of (2.1) are identities of $A$.

The next statement easily follows from Remark 2.1.

Remark 2.2. Suppose that an algebra $A$ satisfies all multilinear identities of an algebra $B$ of degree $\deg f = k\le n$ for some fixed $n$. Then $A^\sharp$ satisfies all identities of $B^\sharp$ of degree $k\le n$.

Using results of [13], we obtain the following inequalities.

Lemma 2.1. ([13,Theorem 2]) Let $A$ be an algebra with an exponentally bounded codimension growth. Then $\overline{exp}(A^\sharp)\le \overline{exp}(A)+1$.

Lemma 2.2. ([13,Theorem 3]) Let $A$ be an algebra with an exponentally bounded codimension growth satisfying the identity (2.2). Then $\underline{exp}(A^\sharp)\ge \underline{exp}(A)+1$.

Given an integer $T\ge 2$, we define an infinite-dimensional algebra $B_T$ by its basis

and by the multiplication table

for all $i\ge 1$ and

All other products of basis elements are equal to zero. Clearly, algebra $B_T$ is right nilpotent of class 3, that is

is an identity of $B_T$. Due to (2.2), any nonzero product of elements of $B_T$ must be left-normed. Therefore we omit brackets in the left-normed products and write $(y_1y_2)y_3 = y_1y_2y_3$ and $(y_1\cdots y_k)y_{k+1} = y_1\cdots y_{k+1}$ if $k\ge 3$.

We will use the following properties of algebra $B_T$.

Lemma 2.3. ([14,Lema 2.1]) Let $n\le T$. Then $c_n(B_T)\le 2n^3$.

Lemma 2.4. ([14,Lema 2.2]) Let $n = kT+1$. Then

Lemma 2.5. ([14,Lema 2.3]) Any multilinear identity $f = f(x_1,\ldots,x_n)$ of degree $n\le T$ of algebra $B_T$ is an identity of $B_{T+1}$.

Let $F[\theta]$ be a polynomial ring over $F$ on one indeterminate $\theta$ and let $F[\theta]_0$ be its subring of all polynomials without free term. Denote by $Q_N$ the quotient algebra

where $(Q^{N+1})$ is an ideal of $F[\theta]$ generated by $Q^{N+1}$. Fix an infinite sequence of integers $T_1<N_1<T_2<N_2\ldots$ and consider the algebra

where $B(T,N) = B_T\otimes Q_N$.

Let $R$ be an algebra of the type (2.3). Then the following lemma holds.

Lemma 2.6. For any $i\ge 1$, the following equalities hold:

(a) if $T_i\le n\le N_i$ then

(b) if $N_i <n\le T_{i+1}$ then

Proof. This follows immediately from the equality $B(T_i,N_i)^{N_i+1} = 0$ and from Lemma 2.5.

The folowing remark is obvious.

Remark 2.3. Ler $R$ be an algebra of type (2.3). Then

3.   The main result

Theorem 3.1. For any real $\alpha>1$, there exists an algebra $R_\alpha$ with $\underline{exp}(R_\alpha) = 1, \overline{exp}(R_\alpha) = \alpha$ such that $\underline{exp}(R_\alpha^\sharp) = 2$ and $\alpha\le\overline{exp}(R_\alpha^\sharp)\le\alpha+1$.

Proof. Note that

for any algebra $A$ satisfying (2.2). We will construct $R_\alpha$ of type (2.3) by a special choice of the sequence $T_1,N_1,T_2,N_2,\ldots$ depending on $\alpha$. First, choose $T_1$ such that

for all $m\ge T_1$. By Lemma 2.4, algebra $B_{T_1}$ has an overexponential codimenson growth. Hence there exists $N_1> T_1$ such that

Consider an arbitrary $n>N_1$. By Remark 2.1, we have

where

By Lemma 2.6, we have $\Sigma_1'+\Sigma_2'\le \Sigma_1+\Sigma_2$, where

Then for any $T_2>N_1$, an upper bound for $\Sigma_2$ is

which follows from (3.2), provided that $n\le T_2$.

Let us find an upper bound for $\Sigma_1$ assuming that $n$ is sufficiently large. Clearly,

which follows from the choice of $N_1$, relation (3.1), and the equality $B(T_1,N_1)^n = 0$ for all $n\ge N_1+1$. Since $N_1\alpha^{N_1}$ is a constant for fixed $N_1$, we only need to estimate the sum of binomial coefficients.

From the Stirling formula

it follows that

Now we define the function $\Phi:[0;1]\to\mathbb R$ by setting

It is not difficult to show that $\Phi$ increases on $[0;1/2]$, $\Phi(0) = 1$, and $\Phi(x)\le 2$ on $[0;1]$. In terms of the function $\Phi$ we rewrite (3.5) as

provided that $n> 2N_1$. Now (3.4) and (3.6) together with (3.3) imply

Since

and $\Phi(x)$ increases on $(0;1/2]$, there exists $n>2N_1$ such that

Now we take $T_2$ to be equal to the minimal $n>2N_1$ satisfying (3.7). Note that for such $T_2$ we have

for $n = T_2$.

As soon as $T_2$ is choosen, we can take $N_2$ as the minimal $n$ such that $c_n(B_{T_2}) \ge \alpha^n$. Then again, $c_m(R)<m \alpha^m$ if $m<N_2$. Repeating this procedure, we can construct an infinite chain $T_1<N_1<T_2<N_2\cdots\;$ such that

for all $n\ne N_1,N_2,\ldots$,

for all $n = N_1,N_2,\ldots$ and

for all $j = 1,2,\ldots\;$.

Let us denote by $R_\alpha$ the just constructed algebra $R$ of type (2.3). Then (3.10) means that

if $n = T_{j+1}, j = 1,2,\ldots\;$. It follows from inequality (3.11) that

On the other hand, since $R_\alpha$ is not nilpotent, it follows that

Since the PI-exponent of non-nilpotent algebra cannot be strictly less than $1$, relations (3.12), (3.13) and Lemma 2.2 imply

Finally, relations (3.8), (3.9) imply the equality $\overline{exp}(R_\alpha) = \alpha$. Applying Lemma 2.1, we see that $\overline{exp}(R_\alpha^\sharp)\le\alpha+1$. The inequality $\alpha = \overline{exp}(R_\alpha)\le \overline{exp}(R_\alpha^\sharp)$ is obvious, since $R_\alpha$ is a subalgebra of $R_\alpha^\sharp$, Thus we have completed the proof of Theorem 3.1.

As a consequence of Theorem 3.1 we get an infinite family of unital algebras of exponential codimension growth without ordinary PI-exponent.

Corollary 1. Let $\beta>2$ be an arbitrary real number. Then the ordinary PI-exponent of unital algebra $R_\beta^\sharp$ from Theorem 3.1 does not exist. Moreover, $\underline{exp}(R_\beta^\sharp) = 2$, whereas $\beta\le\overline{exp}(R_\beta^\sharp)\le\beta+1$.

Acknowledgments

We would like to thank the referee for comments and suggestions.