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Trace codimensions of algebras and their exponential growth

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Abstract

The trace codimensions give a quantitative description of the identities satisfied by an algebra with trace. Here we study the asymptotic behaviour of the sequence of trace codimensions c trn (A) and of pure trace codimensions c ptrn (A) of a finite-dimensional algebra A over a field of characteristic zero. We find an upper and lower bound of both codimensions and as a consequence we get that the limits \({\lim _{n \to \infty}}\,\,\root n \of {c_n^{{\rm{tr}}}\left(A \right)} \) and \({\lim _{n \to \infty}}\,\,\root n \of {c_n^{{\rm{ptr}}}\left(A \right)} \) always exist and are integers. This result gives a positive answer to a conjecture of Amitsur in this setting. Finally we characterize the varieties of algebras whose exponential growth is bounded by 2.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Antonio Giambruno

  2. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio 1, 67100, L’Aquila, Italy

    Antonio Ioppolo

  3. Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Daniela La Mattina

Authors
  1. Antonio Giambruno
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  2. Antonio Ioppolo
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  3. Daniela La Mattina
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Correspondence to Antonio Giambruno.

Additional information

The authors were partially supported by GNSAGA of INdAM.

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Giambruno, A., Ioppolo, A. & La Mattina, D. Trace codimensions of algebras and their exponential growth. Isr. J. Math. 254, 431–459 (2023). https://doi.org/10.1007/s11856-022-2414-3

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  • DOI: https://doi.org/10.1007/s11856-022-2414-3

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