Varieties of associative algebras with slow central codimension growth

Abstract

Central codimensions measure the dimensions of spaces of multilinear polynomials modulo those taking values in the center of an algebra. We classify associative algebras over a field of characteristic zero whose central codimensions grow at most linearly. The classification is given up to equality of polynomial identities, in terms of explicit generating algebras and an equivalent finite exclusion criterion. We also give a direct structural characterization of the bounded case. As a consequence, we determine the five varieties of quadratic central codimension growth whose proper subvarieties all have at most linear central growth. As an application of the bounded-growth classification, we characterize algebras graded by an arbitrary finite group whose graded central codimensions are bounded. Up to graded PI-equivalence, these are direct sums of commutative and nilpotent graded algebras, possibly together with the Grassmann algebra on two generators endowed with the trivial grading.

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BibTeX RIS

BibTeX

@misc{ioppolo2026varieties,
  title = {{Varieties of associative algebras with slow central codimension growth}},
  author = {Antonio Ioppolo and Wesley Quaresma Cota and Ana Cristina Vieira},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01244},
  eprint = {2611.01244},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01244},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Varieties of associative algebras with slow central codimension growth
AU  - Antonio Ioppolo
AU  - Wesley Quaresma Cota
AU  - Ana Cristina Vieira
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01244
UR  - https://arxiv.org/abs/2611.01244
AB  - Central codimensions measure the dimensions of spaces of multilinear polynomials modulo those taking values in the center of an algebra. We classify associative algebras over a field of characteristic zero whose central codimensions grow at most linearly. The classification is given up to equality of polynomial identities, in terms of explicit generating algebras and an equivalent finite exclusion criterion. We also give a direct structural characterization of the bounded case. As a consequence, we determine the five varieties of quadratic central codimension growth whose proper subvarieties all have at most linear central growth. As an application of the bounded-growth classification, we characterize algebras graded by an arbitrary finite group whose graded central codimensions are bounded. Up to graded PI-equivalence, these are direct sums of commutative and nilpotent graded algebras, possibly together with the Grassmann algebra on two generators endowed with the trivial grading.
ER  -

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