Older adults’ activity on a geriatric hospital unit: A behavioral mapping study

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.