On existence of PI-exponents of unital algebras

We construct a family of unital non-associative algebras $\{T_\alpha\vert~ 2<\alpha\in\mathbb R\}$ such that $\underline{exp}(T_\alpha)=2$, whereas $\alpha\le\overline{exp}(T_\alpha)\le\alpha+1$. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $T_\alpha$ does not exist for any $\alpha>2$. This is the first example of a unital algebra whose PI-exponent does not exist.


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, . . ., 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 . At the end of 1980's, Amitsur conjectured that for any associative PI-algebra, the limit (1.1) exists and is a nonnegative integer. Amitsur's conjecture was confirmed in [5,6]. Later, Amitsur's conjecture was also confirmed for finite-dimensional Lie and Jordan algebras [4,13]. 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 α > 1, an algebra R α was constructed such that the lower limit of n c n (A) is equal to 1, whereas the upper limit is equal to α. 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 ♯ the algebra obtained from A by adjoining the external unit. There is a number of papers where the existence of exp(A ♯ ) has been proved, provided that exp(A) exists [11,15,16]. Moreover, in all these cases, exp(A ♯ ) = 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].

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. In [14], an algebra R = R(α) such that exp(R) = 1, exp(R) = α, was constructed for any real α > 0. Slightly modifying the construction from [14], we want to get for any real α > 2, an algebra R α with exp(R α ) ♯ = 2 and α ≤ exp(R ♯ ) ≤ α + 1.
Clearly, polynomial identities of A ♯ strongly depend on the identities of A. In particular, we make the following observation.
where f i1,...,i k is a multilinear polynomial on x i1 , . . . , x i k obtained from f by replacing all x j , j = i 1 , . . . , i k with 1.
and only if all of its components f i1,...,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 ≤ n for some fixed n. Then A ♯ satisfies all identities of B ♯ of degree k ≤ n.
Using results of [17], we obtain the following inequalities. Given an integer T ≥ 2, we define an infinite-dimensional algebra B T by its basis {a, b, z i 1 , . . . , z i T | i = 1, 2, . . .} and by the multiplication table 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 1 y 2 )y 3 = y 1 y 2 y 3 and (y 1 · · · y k )y k+1 = y 1 · · · y k+1 if k ≥ 3. We will use the following properties of algebra B T . Let F [θ] be a polynomial ring over F on one indeterminate θ and let F [θ] 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 [θ] generated by Q N +1 . Fix an infinite sequence of integers T 1 < N 1 < T 2 < N 2 . . . and consider the algebra Let R be an algebra of the type (2.3). Then the following lemma holds.
Lemma 2.6. For any i ≥ 1, the following equalities hold: Proof. This follows immediately from the equality B(T i , N i ) Ni+1 = 0 and from Lemma 2.5.
The folowing remark is obvious.

The main result
Theorem 3.1. For any real α > 1, there exists an algebra R α with exp(R α ) = 1, exp(R α ) = α such that exp(R ♯ α ) = 2 and α ≤ exp(R ♯ α ) ≤ α + 1. Proof. Note that for any algebra A satisfying (2.2). We will construct R α of type (2.3) by a special choice of the sequence T 1 , N 1 , T 2 , N 2 , . . . depending on α. First, choose T 1 such that for all m ≥ T 1 . By Lemma 2.4, algebra B T1 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 By Lemma 2.6, we have Σ ′ Then for any T 2 > N 1 , an upper bound for Σ 2 is which follows from (3.2), provided that n ≤ T 2 . Let us find an upper bound for Σ 1 assuming that n is sufficiently large. Clearly, n k which follows from the choice of N 1 , relation (3.1), and the equality B(T 1 , N 1 ) n = 0 for all n ≥ N 1 + 1. Since N 1 α N1 is a constant for fixed N 1 , we only need to estimate the sum of binomial coefficients.
As a consequence of Theorem 3.1 we get an infinite family of unital algebras of exponential codimension growth without ordinary PI-exponent.