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A Characterization of Superalgebras with Pseudoinvolution of Exponent 2

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Abstract

Let A be a superalgebra endowed with a pseudoinvolution ∗ over an algebraically closed field of characteristic zero. If A satisfies an ordinary non-trivial identity, then its graded ∗-codimension sequence \(c_{n}^{*}(A)\), n = 1,2,…, is exponentially bounded (Ioppolo and Martino (Linear Multilinear Algebra 66(11), 2286–2304 2018). In this paper we capture this exponential growth giving a positive answer to the Amitsur’s conjecture for this kind of algebras. More precisely, we shall see that the \(\lim _{n \rightarrow \infty } \sqrt [n]{c_{n}^{*}(A)}\) exists and it is an integer, denoted \(\exp ^{*}(A)\) and called graded ∗-exponent of A. Moreover, we shall characterize superalgebras with pseudoinvolution according to their graded ∗-exponent.

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References

  1. Aljadeff, E., Giambruno, A.: Multialternating graded polynomials and growth of polynomial identities. Proc. Amer. Math Soc. 141, 3055–3065 (2013)

    Article  MathSciNet  Google Scholar 

  2. Aljadeff, E., Giambruno, A., La Mattina, D.: Graded polynomial identities and exponential growth. J. Reine Angew. Math. 650, 83–100 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Benanti, F., Giambruno, A., Pipitone, M.: Polynomial identities on superalgebras and exponential growth. J. Algebra 269, 422–438 (2003)

    Article  MathSciNet  Google Scholar 

  4. Giambruno, A., Ioppolo, A., La Mattina, D.: Varieties of algebras with superinvolution of almost polynomial growth. Algebr. Represent. Theory 19(3), 599–611 (2016)

    Article  MathSciNet  Google Scholar 

  5. Giambruno, A., Ioppolo, A., La Mattina, D.: Superalgebras with involution or superinvolution and almost polynomial growth of the codimensions. Algebr Represent. Theory 22(4), 961–976 (2019)

    Article  MathSciNet  Google Scholar 

  6. Giambruno, A., La Mattina, D.: Graded polynomial identities and codimensions: Computing the exponential growth. Adv. Math. 225, 859–881 (2010)

    Article  MathSciNet  Google Scholar 

  7. Giambruno, A., Polcino Milies, C., Valenti, A.: Star-polynomial identities: computing the exponential growth of the codimensions. J. Algebra 469, 302–322 (2017)

    Article  MathSciNet  Google Scholar 

  8. Giambruno, A., Zaicev, M.: On codimension growth of finitely generated associative algebras. Adv. Math. 140, 145–155 (1998)

    Article  MathSciNet  Google Scholar 

  9. Giambruno, A., Zaicev, M.: Exponential codimension growth of PI-algebras: an exact estimate. Adv. Math. 142, 221–243 (1999)

    Article  MathSciNet  Google Scholar 

  10. Giambruno, A., Zaicev, M.: Involutions codimensions of finite dimensional algebras and exponential growth. J. Algebra 222, 474–484 (1999)

    Article  MathSciNet  Google Scholar 

  11. Giambruno, A., Zaicev, M.: A characterization of varieties of associative algebras of exponent two. Serdica Math. J. 26, 245–252 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Gordienko, A.S.: Amitsurs’ conjecture for associative algebras with a generalised Hopf action. J. Pure Appl. Algebra 217, 1395–1411 (2013)

    Article  MathSciNet  Google Scholar 

  13. Ioppolo, A.: The exponent for superalgebras with superinvolution. Linear Algebra Appl. 555, 1–20 (2018)

    Article  MathSciNet  Google Scholar 

  14. Ioppolo, A., Martino, F.: Varieties of algebras with pseudoinvolution and polynomial growth. Linear Multilinear Algebra 66(11), 2286–2304 (2018)

    Article  MathSciNet  Google Scholar 

  15. Ioppolo, A., Martino, F.: Classifying G-graded algebras of exponent two. Israel J. Math. 229, 341–356 (2019)

    Article  MathSciNet  Google Scholar 

  16. Jaber, A.: Existence of pseudo-superinvolutions of the first kind, Int. J. Math. Math. Sci. (2008)

  17. Martinez, C., Zelmanov, E.: Representation theory of Jordan superalgebras. I, Trans. Amer. Math. Soc. 362(2), 815–846 (2010)

    Article  MathSciNet  Google Scholar 

  18. Pipitone, M.: Algebras with involution whose exponent of ∗-codimensions is equal to two. Comm. Algebra 30, 3875–3883 (2002)

    Article  MathSciNet  Google Scholar 

  19. Regev, A.: Existence of identities in A ⊗ B. Israel J. Math. 11, 131–152 (1972)

    Article  MathSciNet  Google Scholar 

  20. dos Santos, R.B.: ∗-Superalgebras and exponential growth. J. Algebra 473, 283–306 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author wish to express his thanks to Efim Zelmanov for suggesting the study of superalgebras with pseudoinvolution.

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Authors and Affiliations

  1. IMECC, UNICAMP, Sérgio Buarque de Holanda 651, 13083-859, Campinas, SP, Brazil

    Antonio Ioppolo

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  1. Antonio Ioppolo
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Correspondence to Antonio Ioppolo.

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Presented by: Kenneth Goodearl

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A. Ioppolo was supported by the Fapesp post-doctoral grant number 2018/17464-3

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Ioppolo, A. A Characterization of Superalgebras with Pseudoinvolution of Exponent 2. Algebr Represent Theor 24, 1415–1429 (2021). https://doi.org/10.1007/s10468-020-09996-4

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