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Vol. 51 No. 3–4 (2025)
Serdica Mathematical Journal

Surveys

Minimal varieties of PI-algebras with graded involution: a survey

https://doi.org/10.55630/serdica.2025.51.231-252

Published 23-12-2025

  • Francesca Saviella Benanti
    • ,
  • Onofrio M. Di Vincenzo
    • ,
  • Angela Valenti

Francesca Saviella Benanti
Universitá di Palermo

Onofrio M. Di Vincenzo
Universiá degli Studi della Basilicata

Angela Valenti
Universitá di Palermo

Abstract

This survey provides an overview of recent developments in the structure and classification of minimal varieties of associative PI-algebras equipped with graded involutions, particularly over an algebraically closed field of characteristic zero and under the action of a cyclic group of prime order.

2020 Mathematics Subject Classification:

16R10, 16R50, 16W50, 16P90

Keywords

graded algebras, graded involutions, codimension, minimal varieties

Full text

Author Details

Francesca Saviella Benanti

Dipartimento di Matematica e Informatica
Università di Palermo
via Archirafi, 34
90123 Palermo, Italy
e-mail: francescasaviella.benanti@unipa.it

Onofrio M. Di Vincenzo

Dipartimento di Matematica, Informatica ed Economia
Università degli Studi della Basilicata
via dell’Ateneo Lucano, 10
85100 Potenza, Italy
e-mail: onofrio.divincenzo@unibas.it

Angela Valenti

Dipartimento di Ingegneria
Università di Palermo
Viale delle Scienze
90128 Palermo, Italy
e-mail: angela.valenti@unipa.it

References

  1. E. Aljadeff, A. Giambruno. Multialternating graded polynomials and growth of polynomial identities. Proc. Amer. Math. Soc. 141, 9 (2013), 3055–3065.
  2. E. Aljadeff, A. Giambruno, Y. Karasik. Polynomial identities with involution, superinvolutions and the Grassmann envelope. Proc. Amer. Math. Soc. 145, 5 (2017), 1843–1857.
  3. S. Argenti, O. M. Di Vincenzo. Minimal varieties of graded PI-algebras over abelian groups. Bull. Lond. Math. Soc. 56, 7 (2024), 2441–2459.
  4. Yu. A. Bahturin, M. Kochetov. Classification of group gradings on simple Lie algebras of types A, B, C and D. J. Algebra 324, 6 (2010), 2971–2989.
  5. Yu. A. Bahturin, M. V. Zaicev, S. K. Sehgal. Finite-dimensional simple graded algebras. Mat. Sb. 199, 7 (2008), 21–40 (in Russian); English translation in: Sb. Math. 199, 7–8 (2008), 965–983
  6. Yu. A. Bahturin, I. P. Shestakov, M. V. Zaicev. Gradings on simple Jordan and Lie algebras. J. Algebra 283, 2 (2005), 849–868.
  7. Yu. A. Bahturin, M. V. Zaicev. Involutions on graded matrix algebras. J. Algebra 315, 2 (2007), 527–540.
  8. F. S. Benanti, A. Giambruno, M. Pipitone. Polynomial identities on superalgebras and exponentil growth. J. Algebra 269, 2 (2003), 422–428.
  9. F. S. Benanti, O. M. Di Vincenzo, A. Valenti. Minimal varieties of PI-algebras with graded involution. Linear Algebra Appl. 699 (2024), 459–507.
  10. O. M. Di Vincenzo, R. La Scala. Minimal algebras with respect to their *-exponent. J. Algebra 317, 2 (2007), 642–657.
  11. O. M. Di Vincenzo, V. C. Nardozza. Minimal affine varieties of superalgebras with superinvolution: a characterization. Linear Algebra Appl. 698 (2024), 514–536.
  12. O. M. Di Vincenzo, E. Spinelli. A characterization of *-minimal algebras with involution. Israel J. Math. 186 (2011), 381–400.
  13. O. M. Di Vincenzo, E. Spinelli. Graded polynomial identities on upper block triangular matrix algebras. J. Algebra 415 (2014), 50–64.
  14. O. M. Di Vincenzo, V. R. T. da Silva, E. Spinelli. A characterization of minimal varieties of Zp-graded PI algebras. J. Algebra 539 (2019), 397–418.
  15. O. M. Di Vincenzo, V. R. T. da Silva, E. Spinelli. Minimal varieties of PI-superalgebras with graded involution. Israel J. Math. 241, 2 (2021), 869–909.
  16. O. M. Di Vincenzo, A. Valenti. ∗-multilinear polynomials with invertible values. Rend. Sem. Mat. Univ. Padova 86 (1991), 47–55.
  17. A. Giambruno, A. Ioppolo, D. La Mattina. Superalgebras with involution or superinvolution and almost polynomial growth of the codimensions. Algebr. Represent. Theory 22, 4 (2019), 961–976.
  18. A. Giambruno, D. La Mattina, C. P. Milies. Star-fundamental algebras: polynomial identities and asymptotics. Trans. Amer. Math. Soc. 373, 11 (2020), 7869–7899.
  19. A. Giambruno, C. P. Milies, A. Valenti. Star-polynomial identities: Computing the exponential growth of the codimensions. J. Algebra 469 (2017), 302–322.
  20. A. Giambruno, M. Zaicev. On codimension growth of finitely generated associative algebras. Adv. Math. 140, 2 (1998), 145–155.
  21. A. Giambruno, M. Zaicev. Exponential codimension growth of PI algebras: an exact estimate. Adv. Math. 142, 2 (1999), 221–243.
  22. A. Giambruno, M. Zaicev. Involution codimension of finite-dimensional algebras and exponential growth. J. Algebra 222, 2 (1999), 471–484.
  23. A. Giambruno, M. Zaicev. Minimal varieties of algebras of exponential growth. Adv. Math. 174, 2 (2003), 310–323.
  24. A. Giambruno, M. Zaicev. Codimension growth and minimal superalgebras. Trans. Amer. Math. Soc. 355, 12 (2003), 5091–5117.
  25. A. Giambruno, M. Zaicev. Polynomial Identities and Asymptotic Methods. Math. Surveys Monogr., vol. 122. Providence, RI, AMS, 2005.
  26. A. S. Gordienko. Amitsur’s conjecture for associative algebras with a generalized Hopf action. J. Pure Appl. Algebra 217, 8 (2013), 1395–1411.
  27. A. S. Gordienko. Asymptotics of H-identities for associative algebras with an H-invariant radical. J. Algebra 393 (2013), 92–101.
  28. A. Ioppolo. The exponent for superalgebras with superinvolution. Linear Algebra Appl. 555 (2018), 1–20.
  29. V. G. Kac. Lie superalgebras. Advances in Math. 26, 1 (1977), 8–96.
  30. A. R. Kemer. Ideals of identities of associative algebras. Transl. Math. Monogr., vol. 87. Providence, RI, Amer. Math. Soc., 1991.
  31. M. L. Racine, E. I. Zel’manov. Simple Jordan superalgebras with semisimple even part. J. Algebra 270, 2 (2003), 374–444.
  32. A. Regev. Existence of identities in A ⊗ B. Israel J. Math. 11 (1972), 131–152.
  33. L. H. Rowen. Polynomial Identities in Ring Theory. Pure Appl. Math., vol. 84. New York-London, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], 1980.
  34. I. Sviridova. Finitely generated algebras with involution and their identities. J. Algebra 383 (2013), 144–167.