## Abstract

Let *F* be a field of characteristic zero and let \(\mathcal{V}\) be a variety of associative *F*-algebras graded by a finite abelian group *G*. If \(\mathcal{V}\) satisfies an ordinary non-trivial identity, then the sequence \(c_n^G(\mathcal{V})\) of *G*-codimensions is exponentially bounded. In [2, 3, 8], the authors captured such exponential growth by proving that the limit \(^G(\mathcal{V}) = {\rm{lim}}_{n \to \infty} \sqrt[n]{{c_n^G(\mathcal{V})}}\)
exists and it is an integer, called the *G*-exponent of \(\mathcal{V}\).

The purpose of this paper is to characterize the varieties of *G*-graded algebras of exponent greater than 2. As a consequence, we find a characterization for the varieties with exponent equal to 2.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]E. Aljadeff and A. Kanel-Belov,
*Representability and Specht problem for G-graded algebras*, Advances in Mathematics**226**(2010), 2391–2428.MathSciNetCrossRefGoogle Scholar - [2]E. Aljadeff and A. Giambruno,
*Multialternating graded polynomials and growth of polynomial identities*, Proceedings of the American Mathematical Society**141**(2013), 3055–3065.MathSciNetCrossRefGoogle Scholar - [3]E. Aljadeff, A. Giambruno and D. La Mattina,
*Graded polynomial identities and exponential growth*, Journal für die Reine und Angewandte Mathematik**650**(2011), 83–100.MathSciNetzbMATHGoogle Scholar - [4]Y. A. Bahturin, S. K. Sehgal and M. V. Zaicev,
*Finite-dimensional simple graded algebras*, Sbornik. Mathematics**199**(2008), 965–983.MathSciNetCrossRefGoogle Scholar - [5]F. Benanti, A. Giambruno and M. Pipitone,
*Polynomial identities on superalgebras and exponential growth*, Journal of Algebra**269**(2003), 422–438.MathSciNetCrossRefGoogle Scholar - [6]M. Cohen and S. Montgomery,
*Group-graded rings, smash product and group actions*, Transactions of the American Mathematical Society**282**(1984), 237–258.MathSciNetCrossRefGoogle Scholar - [7]C. W. Curtis and I. Reiner,
*Representation Theory of Finite Groups and Associative Algebras*, Wiley Classics Library, John Wiley & Sons, New York, 1988.zbMATHGoogle Scholar - [8]A. Giambruno and D. La Mattina,
*Graded polynomial identities and codimensions: Computing the exponential growth*, Advances in Mathematics**225**(2010), 859–881.MathSciNetCrossRefGoogle Scholar - [9]A. Giambruno, C. Polcino Milies and A. Valenti,
*Star-polynomial identities: computing the exponential growth of the codimensions*, Journal of Algebra**469**(2017), 302–322.MathSciNetCrossRefGoogle Scholar - [10]A. Giambruno and A. Regev,
*Wreath products and P.I. algebras*, Journal of Pure and Applied Algebra**35**(1985), 133–149.MathSciNetCrossRefGoogle Scholar - [11]A. Giambruno and M. Zaicev,
*On codimension growth of finitely generated associative algebras*, Advances in Mathematics**140**(1998), 145–155.MathSciNetCrossRefGoogle Scholar - [12]A. Giambruno and M. Zaicev,
*Exponential codimension growth of PI-algebras: an exact estimate*, Advances in Mathematics**142**(1999), 221–243.MathSciNetCrossRefGoogle Scholar - [13]A. Giambruno and M. Zaicev,
*Involutions codimensions of finite dimensional algebras and exponential growth*, Journal of Algebra**222**(1999), 474–484.MathSciNetCrossRefGoogle Scholar - [14]A. Giambruno and M. Zaicev,
*A characterization of varieties of associative algebras of exponent two*, Serdica**26**(2000), 245–252.MathSciNetzbMATHGoogle Scholar - [15]A. S. Gordienko,
*Amitsurs’ conjecture for associative algebras with a generalised Hopf action*, Journal of Pure and Applied Algebra**217**(2013), 1395–1411.MathSciNetCrossRefGoogle Scholar - [16]A. Ioppolo,
*The exponent for superalgebras with superinvolution*, Linear Algebra and its Applications**555**(2018), 1–20.MathSciNetCrossRefGoogle Scholar - [17]M. Pipitone,
*Algebras with involution whose exponent of-codimensions is equal to two*, Communications in Algebra**30**(2002), 3875–3883.MathSciNetCrossRefGoogle Scholar - [18]A. Regev,
*Existence of identities in A B*, Israel Journal of Mathematics**11**(1972), 131–152.MathSciNetCrossRefGoogle Scholar - [19]R. B. dos Santos, -Superalgebras and exponential growth, Journal of Algebra
**473**(2017), 283–306.Google Scholar - [20]D. Stefan and F. Van Oystaeyen,
*The Wedderburn–Malcev theorem for comodule algebras*, Communications in Algebra**27**(1999), 3569–3581.MathSciNetCrossRefGoogle Scholar - [21]I. Sviridova,
*Identities of pi-algebras graded by a finite abelian group*, Communications in Algebra**39**(2011), 3462–3490.MathSciNetCrossRefGoogle Scholar - [22]A. Valenti,
*Group graded algebras and almost polynomial growth*, Journal of Algebra**334**(2011), 247–254.MathSciNetCrossRefGoogle Scholar