## ABSTRACT

The polynomial identities satisfied by a given algebra A over a field of characteristic zero can be measured through the sequence of codimensions cn(A), n = 1, 2, . . . , of the algebra. This sequence was introduced by Regev in [21] and its n-th term measures the dimension of the space of multilinear polynomials in n variables in the corresponding relatively free algebra of countable rank. Since in characteristic zero, by the multilinearization process, every identity is equivalent to a system of multilinear ones, the sequence of codimensions of A gives a quantitative measure of the identities satisfied by the given algebra. Maybe the most important feature of this sequence proved in [21] is that in case A is an associative algebra satisfying a polynomial identity (PI-algebra), then cn(A) is exponentially bounded. A primary tool for studying the identities of an algebra is provided by

the representation theory of the symmetric group Sn. By the above reasons and also since the representation theory of Sn is well developed in characteristic zero, most attention has been focused on algebras over a field of characteristic zero and we shall make such assumption here. The exponential rate of growth of the sequence of codimensions was de-

termined in [9] and [10] by Giambruno and Zaicev who proved that for any

PI-algebra A, the limit limn→∞ n √ cn(A) exists and is a nonnegative integer.

Such integer denoted exp(A) is called the PI-exponent of the algebra A. Having at hand a scale provided by the PI-exponent most work has been concentrated in recent years in trying to characterize PI-algebras, or their identities, having the same PI-exponent (see [13], [14], [12]). Here we shall be dealing with algebras A for which exp(A) ≤ 1, i.e., whose

sequence of codimensions is polynomially bounded. In [16], [17] Kemer gave a characterization of the algebras (or of their identities) whose sequence of codimensions is polynomially bounded. He proved that A is such an algebra if and only if G and UT2 do not satisfy all the identities of the algebra A, where G is the Grassmann algebra and UT2 is the algebra of 2 × 2 upper triangular matrices. As a consequence of this result he observed that the sequence of codimensions of any PI-algebra A, is either polynomially bounded or grows exponentially. Hence no intermediate growth is allowed. A refinement of the result of Kemer was recently achieved in [4]. In

fact, there the authors completely classified all T-ideals whose sequence of codimensions is linearly bounded. As a consequence it turned out that the allowed linear functions can be classified. The purpose of this paper is to present these results and a generalization

in the setting of superalgebras. Since the ordinary algebras are superalgebras with trivial grading we shall be able to recover the classification obtained in [4]. We should mention that the superalgebras and their graded identities play

a relevant role in the structure theory of varieties developed by Kemer (see [18]). In the case of superalgebras, one defines similar invariants measuring the growth of the graded identities ([8]). In particular in the sequence of graded codimensions cgrn (A), n = 1, 2, . . . , of a superalgebra A, the n-th term measures the dimension of the space of multilinear elements of the relatively free superalgebra of countable rank of A. It turns out that if a superalgebra satisfies an ordinary identity, then its sequence of graded codimensions is exponentially bounded ([8]). Moreover the hyperoctahedral group Z2 o Sn and its representation theory are a natural tool for studying the graded identities of a superalgebra in characteristic zero. The problem of characterizing the graded identities of a superalgebra

whose sequence of graded codimensions is polynomially bounded was studied in [7]. It was proved that a superalgebra A has such property if and only if its graded identities are not a consequence of the graded identities of five explicit superalgebras. Four of these algebras are the algebras G and UT2 endowed with suitable Z2-gradings. In particular these results show that also for the superalgebras no intermediate growth is allowed. For finitely generated superalgebras satisfying an ordinary polynomial

identity, it was shown in [1] that the limit limn→∞ n √ cgrn (A) exists and is

a nonnegative integer, and a theory extending the ordinary case is being developed. Here we shall present a complete classification of the ideals of identities

and graded identities whose sequence of codimensions and graded codimensions is linearly bounded. Moreover for each such ideal I we shall exhibit an algebra or superalgebra having I as ideal of identities or graded identities.

2. Generalities