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Growth functions of algebras and an application to Leavitt path algebras

  • João Schwarz
  • Alfilgen Sebandal
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Abstract

In this paper, we consider three growth functions of algebras: the Gelfand-Kirillov dimension, superdimension, and entropy -- as well as a variation of the latter successfully used in the study of Leavitt path algebras. We prove results that make precise the heuristic fact that the Gelfand-Kirillov dimension is suitable for the study of algebras with polynomial growth, the superdimension for algebras with subexponential growth, and the entropy for algebras with exponential growth. We introduce a definition of entropy for graded modules motivated by the entropy for graded algebras. As an application of our study, we give a new criterion for Leavitt path algebras to be PI.

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Cite

BibTeXRIS
BibTeX
@misc{schwarz2026growth,
  title = {{Growth functions of algebras and an application to Leavitt path algebras}},
  author = {João Schwarz and Alfilgen Sebandal},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18144},
  eprint = {2609.18144},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18144},
}
RIS
TY  - UNPB
TI  - Growth functions of algebras and an application to Leavitt path algebras
AU  - João Schwarz
AU  - Alfilgen Sebandal
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18144
UR  - https://arxiv.org/abs/2609.18144
AB  - In this paper, we consider three growth functions of algebras: the Gelfand-Kirillov dimension, superdimension, and entropy -- as well as a variation of the latter successfully used in the study of Leavitt path algebras. We prove results that make precise the heuristic fact that the Gelfand-Kirillov dimension is suitable for the study of algebras with polynomial growth, the superdimension for algebras with subexponential growth, and the entropy for algebras with exponential growth. We introduce a definition of entropy for graded modules motivated by the entropy for graded algebras. As an application of our study, we give a new criterion for Leavitt path algebras to be PI.
ER  -