ON EXISTENCE OF PI-EXPONENTS OF CODIMENSION GROWTH

. We construct a family of examples of non-associative algebras { R α | 1 < α ∈ R } such that exp( R α ) = 1, exp( R α ) = α . In particular, it follows that for any R α , an ordinary PI-exponent of codimension growth does not exist.

For a wide class of algebras, the sequence {c n (A)} is bounded by exponential functions a n . This class contains all associative PI-algebras [23], all finite dimensional algebras [2], Kac-Moody algebras [26], infinite dimensional simple Lie algebras of Cartan type [18], and many others. Clearly, the inequality c n (A) ≤ a n implies an existence of upper and lower limits lim sup  called upper and lower PI-exponents of A, respectively. If an ordinary limit of n c n (A) exists, that is, if exp(A) = exp(A), it is called (an ordinary) PI-exponent of A.
One of the main problems of the theory of numerical invariants of polynomial identities is the problem of existence of PI-exponent. At the end of 80's Amitsur conjectured that for any associative algebra with a non-trivial polynomial identity PI-exponent exists and it is a non-negative integer. Amitsur's conjecture was confirmed in [10].
In Lie case existence and integrality of PI-exponent were proved for all finite dimensional algebras [27] and for some classes of infinite dimensional algebras (see, for example, [19]).
Up to now there was no example of an algebra A with exp(A) = exp(A). The main result of our paper is the following theorem. All details about polynomial identities and their numerical characteristics one can find in [1], [7], [11].

Main definitions and constructions
Let F be a field of characteristic zero and let A be an algebra over F . Denote by F {X} the absolutely free algebra over F with the countable set of generators Consider the subspace P n ⊂ F {X} of all multilinear polynomials on x 1 , . . . , x n . Then P n ∩ Id(A) consists of all multilinear identities of A of degree n. It is wellknown that the family of subspaces Denote by The non-negative integer c n (A) is called n th codimension of A. The sequence {c n (A)} is one of the most important numerical characteristics of polynomial identities of A. For proving our main result, we need some intermediate constructions and results. First, given an integer T ≥ 2, we define an algebra B T by its basis and by the multiplication table We suppose that all other products of basis element are equal to zero. It is easy to see that B T is left nilpotent of step 2 algebra, that is, is an identity of B T . From (2.2), it follows that only left-normed products of basis elements may be non-zero. Therefore we will omit brackets in left-normed products of elements of B T . That is we will 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 )y k+1 if k ≥ 3. First, we estimate codimension growth of B T .
where S k is the permutation group, and prove that they are linearly independent modulo the ideal Id(B T ) of identities of B T . Suppose that and, for example, α e = 0, where e is the unit of S k . Then the evaluation ϕ(x 0 ) = z 1 1 ; ϕ(y 1 ) = b 1 , . . . , ϕ(y k ) = b k , ϕ(x q ) = a, q = 1, . . . , k(T − 1), maps f e to z k+1 1 while ϕ(f σ ) = 0 for all σ = e. Hence ϕ(h) = 0, a contradiction. This means that α e = 0 in (2.6). Similarly, all other α σ in (2.6) are equal to zero. Since the total number of elements (2.5) is equal to k! and all f σ lie in P kT +1 , the proof of lemma is completed. Now we compare identities of small degree of algebras B T with distinct T . Proof. Let n ≤ T . It is sufficient to prove that if f = f (x 1 , . . . , x n ) ∈ P n is not an identity of B T +1 then f is not an identity of B T . Let h = σ∈Sn α σ x σ(1) · · · x σ(n) and let ϕ : X → B T +1 be an evaluation such that ϕ(x 1 ), . . . , ϕ(x n ) are basis elements of B ( T + 1) and ϕ(f ) = 0. Then exactly one of x 1 , . . . , x n should be replaced by some z i j while all remaining x k should be replaced by a or b m , m ≥ 1. We may assume that ϕ(x 1 ) = z i j . First let ϕ(x 2 ) = · · · = ϕ(x n ) = a. Then Since ϕ(f ) = 0 then λ = 0 and j + n − 1 ≤ T + 1.
Then there exists an evaluation ϕ : X → B T +1 such that ϕ(x 1 ) = z i j , ϕ(x k ) = b i for some 2 ≤ k ≤ n, ϕ(x r ) = a if r = 1, k and ϕ(f ) = 0. As before, we can assume that k = 2. Then Moreover, p + q = n − 2 and j + p = T + 1. (2.7) From (2.7), it follows that j ≥ 2 and q + 1 ≤ T . Hence in B T and f is not an identity of B T .

Main result
Now we are ready to prove Theorem 1.1.
Proof. Fix a real number α > 1. Denote by R N the quotient algebra where F [Y ] 0 is the ring of polynomials without free term and (Y N +1 ) is its ideal generated by Y N +1 . Then R N N = 0, R N +1 N = 0. Denote also B(T, N ) = B T ⊗ R N . We will construct an algebra A with exp(A) = 1, exp(A) = α as a direct sum (3.1) The sequence T 1 < N 1 < T 2 < N 2 < . . . we will choose during the proof. First note that if T i < n ≤ N i then multilinear identities of A of degree n coincide with identities of by Lemma 2.3. In particular, and by Lemma 2.1. First, we choose T 1 such that the inequality holds for all m ≥ T 1 . By Lemma 2.2, codimension growth of B T1 is overexponential. Hence, one can find N 1 > T 1 such that c n (B T1 ) < α n for all n ≤ N 1 − 1 and c N1 (B T1 ) ≥ α N1 . for all T 1 ≤ n ≤ N 1 − 1 as follows from (3.2), (3.4), (3.5), and from Lemma 2.1.
On the next step we choose N 2 > T 2 satisfying the relations similar to (3.5), (3.6), and (3.7). Continuing this procedure we obtain an infinite sequence T 1 < N 1 < T 2 < N 2 < . . . such that