Abstract
In this paper we prove that if A is any algebra with involution * satisfying a non-trivial polynomial identity, then its sequence of *-codimensions is eventually non-decreasing. Furthermore, by making use of the *-exponent we reconstruct the only two *-algebras, up to T*-equivalence, generating varieties of almost polynomial growth. As a third result we characterize the varieties of algebras with involution whose exponential growth is bounded by 2.
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E. Aljadeff, A Giambruno and Y. Karasik, Polynomial identities with involution, superinvolutions and the Grassmann envelope, Proceedings of the American Mathematical Society 145 (2017), 1843–1857.
Yu. Bahturin, M. Tvalavadze and T. Tvalavadze, Group gradings on superinvolution simple superalgebras, Linear Algebra and its Applications 431 (2009), 1054–1069.
R. Bezerra dos Santos, *-superalgebras and exponential growth, Journal of Algebra 473 (2017), 283–306.
A. Giambruno, A. Ioppolo and D. La Mattina, Varieties of algebras with superinvolution of almost polynomial growth, Algebras and Representation Theory 19 (2016), 599–611
A. Giambruno, A. Ioppolo and D. La Mattina, Superalgebras with involution or superinvolution and almost polynomial growth of the codimensions, Algebras and Representation Theory 22 (2019), 961–976.
A. Giambruno, D. La Mattina and C. Polcino Milies, Star-fundamental algebras: polynomial identities and asymptotics, Transactions of teh American Mathematical Society, to appear, https://doi.org/10.1090/tran/8182.
A. Giambruno and S. Mishchenko, On star-varieties with almost polynomial growth, Algebra Colloquium 8 (2001), 33–42.
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.
A. Giambruno and A. Regev, Wreath products and P.I. algebras, Journal of Pure and Applied Algebra 35 (1985), 133–149.
A. Giambruno and M. Zaicev, Asymptotics for the standard and the Capelli identities, Israel Journal of Mathematics 135 (2003), 125–145.
A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Mathematical Surveys and Monographs, Vol. 122, American Mathematical Society, Providence, RI, 2005.
C. Gomez-Ambrosi and I. P. Shestakov, On the Lie structure of the skew-elements of a simple superalgebra with involution, Journal of Algebra 208 (1998), 43–71.
A. Ioppolo, The exponent for superalgebras with superinvolution, Linear Algebra and its Applications 555 (2018), 1–20.
A. Ioppolo and D. La Mattina, Polynomial codimension growth of algebras with involutions and superinvolutions, Journal of Algebra 472 (2017), 519–545.
D. La Mattina, T. do Nascimento Silva and A. C. Vieira, Minimal star-varieties of polynomial growth and bounded colength, Journal of Pure and Applied Algebra 222 (2018), 1765–1785.
D. La Mattina and F. Martino, Polynomial growth and star-varieties, Journal of Pure and Applied Algebra 220 (2016), 246–262.
D. La Mattina, S. Mauceri and P. Misso, Polynomial growth and identities of superalgebras and star-algebras, Journal of Pure and Applied Algebra 213 (2009), 2087–2094.
D. La Mattina and P. Misso, Algebras with involution with linear codimension growth, Journal of Algebra 305 (2006), 270–291.
S. Mishchenko and A. Valenti, A star-variety with almost polynomial growth, Journal of Algebra 223 (2000), 66–84.
M. L. Racine, Primitive superalgebras with superinvolution, Journal of Algebra 206 (1998), 588–614.
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Partially supported by GNSAGA of INdAM.
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Giambruno, A., La Mattina, D. Codimensions of star-algebras and low exponential growth. Isr. J. Math. 239, 1–20 (2020). https://doi.org/10.1007/s11856-020-2023-y
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DOI: https://doi.org/10.1007/s11856-020-2023-y