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Central Polynomials of Algebras and Their Growth

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Polynomial Identities in Algebras

Part of the book series: Springer INdAM Series ((SINDAMS,volume 44))

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Abstract

A polynomial in noncommutative variables taking central values in an algebra A is called a central polynomial of A. For instance the algebra of k × k matrices has central polynomials. For general algebras the existence of central polynomials is not granted. Nevertheless if an algebra has such polynomials, how can one measure how many are there?

The growth of central polynomials for any algebra satisfying a polynomial identity over a field of characteristic zero was started in recent years and here we shall survey the results so far obtained.

It turns out that one can prove the existence of two limits called the central exponent and the proper central exponent of A. They give a measure of the exponential growth of the central polynomials and the proper central polynomials of any algebra A satisfying a polynomial identity. They are closely related to exp(A), the PI-exponent of the algebra.

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Acknowledgements

The first author “Antonio Giambruno” was partially supported by the GNSAGA of INDAM. The second author “Mikhail Zaicev” was supported by the Russian Science Foundation, grant 16-11-10013.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Palermo, Palermo, Italy

    Antonio Giambruno

  2. Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, Russia

    Mikhail Zaicev

Authors
  1. Antonio Giambruno
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  2. Mikhail Zaicev
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Correspondence to Antonio Giambruno .

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Editors and Affiliations

  1. Dipartimento di Matematica, Informatica ed Economia, Università della Basilicata, Potenza, Italy

    Onofrio Mario Di Vincenzo

  2. Dipartimento di Matematica e Informatica, Università di Palermo, Palermo, Italy

    Antonio Giambruno

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Giambruno, A., Zaicev, M. (2021). Central Polynomials of Algebras and Their Growth. In: Di Vincenzo, O.M., Giambruno, A. (eds) Polynomial Identities in Algebras. Springer INdAM Series, vol 44. Springer, Cham. https://doi.org/10.1007/978-3-030-63111-6_10

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