Hostname: page-component-65b85459fc-4f4fx Total loading time: 0 Render date: 2025-10-17T01:55:28.545Z Has data issue: false hasContentIssue false
  • English
  • Français

On the central exponent of superalgebras with superinvolution

Part of: Rings and algebras with additional structure Rings with polynomial identity Representation theory of groups

Published online by Cambridge University Press:  22 September 2025

Ginevra Giordani ,
Antonio Ioppolo Open the ORCID record for Antonio Ioppolo [Opens in a new window] ,
Antônio dos Santos  and
Ana Vieira
Ginevra Giordani
Affiliation:
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila , Via Vetoio 1, 67100 L’Aquila, Italy e-mail: ginevra.giordani@graduate.univaq.it antonio.ioppolo@univaq.it
Antonio Ioppolo
Affiliation:
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila , Via Vetoio 1, 67100 L’Aquila, Italy e-mail: ginevra.giordani@graduate.univaq.it antonio.ioppolo@univaq.it
Antônio dos Santos
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais , Avenida Antonio Carlos 6627, Belo Horizonte 31123-970, Brazil e-mail: aapds1510@gmail.com
Ana Vieira*
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais , Avenida Antonio Carlos 6627, Belo Horizonte 31123-970, Brazil e-mail: aapds1510@gmail.com
*
e-mail: anacris@ufmg.br
Get access Rights & Permissions [Opens in a new window]

Abstract

The growth of central polynomials for matrix algebras over a field of characteristic zero was first studied by Regev in $2016$. This problem can be generalized by analyzing the behavior of the dimension $c_n^z(A)$ of the space of multilinear polynomials of degree n modulo the central polynomials of an algebra A. In $2018$, Giambruno and Zaicev established the existence of the limit $\lim \limits _{n \to \infty }\sqrt [n]{c_n^{z}(A)}.$ In this article, we extend this framework to superalgebras equipped with a superinvolution, proving both the existence and the finiteness of the corresponding limit.

Keywords

Central polynomialidentitysuperinvolutionexponentgrowth

MSC classification

Primary: 16R10: $T$-ideals, identities, varieties of rings and algebras 16R50: Other kinds of identities (generalized polynomial, rational, involution)
Secondary: 16W10: Rings with involution; Lie, Jordan and other nonassociative structures 16W50: Graded rings and modules 20C30: Representations of finite symmetric groups

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 18
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

G.G. and A.I. were partially supported by INdAM - GNSAGA. A.A.P.d. was partially supported by CAPES. A.C.V. was partially supported by FAPEMIG and by CNPq.

References

Aljadeff, E., Giambruno, A., and Karasik, Y., Polynomial identities with involution, superinvolutions and the Grassmann envelope . Proc. Am. Math. Soc. 145(2017), no. 5, 1843–1857.Google Scholar
Aljadeff, E., Giambruno, A., and La Mattina, D., Graded polynomial identities and exponential growth . J. Reine Angew. Math. 650(2011), 83–100.Google Scholar
Bahturin, Y., Tvalavadze, M., and Tvalavadze, T., Group gradings on superinvolution simple superalgebras . Linear Algebra Appl. 431(2009), 1054–1069.Google Scholar
Centrone, L., Estrada, A., and Ioppolo, A., On PI-algebras with additional structures: Rationality of Hilbert series and Specht’s problem . J. Algebra 592(2022), 300–356.Google Scholar
Costa, W. D. S., Ioppolo, A., dos Santos, R. B., and Vieira, A. C., Unitary superalgebras with graded involution or superinvolution of polynomial growth . J. Pure Appl. Algebra 225(2021), 106666.Google Scholar
Cota, W. Q., La Mattina, D., dos Santos, A. A., and Vieira, A. C., Exponential growth of central codimensions of graded algebras with involution . Isr. J. Math. (2025) (to appear).Google Scholar
DiVincenzo, O. M. and Nardozza, V., Minimal affine varieties of superalgebras with superinvolution: A characterization . Linear Algebra Appl. 698(2024), 514–536.Google Scholar
DiVincenzo, O. M. and Nardozza, V., Minimal $\ast$ -varieties and superinvolutions . Isr. J. Math. 265(2025), 311–346.Google Scholar
dos Santos, A. A., dos Santos, R. B., and Vieira, A. C., Central exponent of algebras with graded involution . Int. J. Algebra Comput. 35(2025), no. 3, 467–487.Google Scholar
Giambruno, A., Ioppolo, A., and LaMattina, D., Varieties of algebras with superinvolution of almost polynomial growth . Algebr. Represent. Theory. 19(2016), no. 3, 599–611.Google Scholar
Giambruno, A. and La Mattina, D., Graded polynomial identities and codimensions: Computing the exponential growth . Adv. Math. 225(2010), 859–881.Google Scholar
Giambruno, A., Polcino Milies, C., and Valenti, A., Star-polynomial identities: Computing the exponential growth of the codimensions . J. Algebra 469(2017), 302–322.Google Scholar
Giambruno, A. and Zaicev, M., Exponential codimension growth of PIalgebras: An exact estimate . Adv. Math. 142(1999), 221–243.Google Scholar
Giambruno, A. and Zaicev, M., Central polynomials of associative algebras and their growth . Proc. Am. Math. Soc. 147(2018), 909–919.Google Scholar
Gomez-Ambrosi, C. and Shestakov, I. P., On the lie structure of the skew-elements of a simple superalgebra with involution . J. Algebra 208(1998), no. 1, 43–71.Google Scholar
Gordienko, A. S., Amitsur’s conjecture for associative algebras with a generalized Hopf action . J. Pure Appl. Algebra 217(2013), 1395–1411.Google Scholar
Ioppolo, A., The exponent for superalgebras with superinvolution . Linear Algebra Appl. 555(2018), 1–20.Google Scholar
Ioppolo, A., Graded linear maps on superalgebras . J. Algebra 605(2022), 377–393.Google Scholar
Ioppolo, A. and Martino, F., On multiplicities of cocharacters for algebras with superinvolution . J. Pure Appl. Algebra 225(2021), 106536.Google Scholar
Kac, V. G., Lie superalgebras . Adv. Math. 26(1977), no. 1, 8–96.Google Scholar
La Mattina, D., dosSantos, R. B., and Vieira, A. C., Proper central exponent of superalgebras with graded involution or superinvolution . Math. Z. 309(2025), 59, 16 pp.Google Scholar
La Mattina, D., Martino, F., and Rizzo, C., Central polynomials of graded algebras: Capturing their exponential growth . J. Algebra. 600(2022), 45–70.Google Scholar
Martino, F. and Rizzo, C., Growth of central polynomials of algebras with involution . Trans. Am. Math. Soc. 375(2022), 429–453.Google Scholar
Racine, M. L., Primitive superalgebras with superinvolution . J. Algebra 206(1998), no. 2, 588–614.Google Scholar
Racine, M. L. and Zelmanov, E. I., Simple Jordan superalgebras with semisimple even part . J. Algebra 270(2003), no. 2, 374–444.Google Scholar
Regev, A., Existence of identities in $A\otimes B$ . Isr. J. Math. 11(1972), 131–152.Google Scholar
Regev, A., Growth of the central polynomials . Commun. Algebra 44(2016), 4411–4421.Google Scholar
Wall, C., Graded Brauer groups . J. Reine Angew. Math. 213(1963), 187–199.Google Scholar