The Fraïssé Limit of Matrix Algebras with the Rank Metric

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Abstract:

We realize the $\mathbb{F}_q$-algebra $M(\mathbb{F}_q)$ studied by von Neumann and Halperin as the Fraïssé limit of the class of finite-dimensional matrix algebras over a finite field $\mathbb{F}_q$ equipped with the rank metric. We then provide a new Fraïssé-theoretic proof of uniqueness of such an object. Using the results of Carderi and Thom, we show that the automorphism group of $\mathrm{Aut}(\mathbb{F}_q )$ is extremely amenable. We deduce a Ramsey-theoretic property for the class of algebras $M(\mathbb{F}_q)$, and provide an explicit bound for the quantities involved.

Citation

Aaron Anderson. 2017. “The Fraïssé Limit of Matrix Algebras with the Rank Metric.” arXiv preprint, https://arxiv.org/abs/1712.04431.

@unpublished{anderson2021fraisselimitmatrixalgebras,
      title={The Fra\"iss\'e limit of matrix algebras with the rank metric}, 
      author={Aaron Anderson},
      year={2021},
      eprint={1712.04431},
      archivePrefix={arXiv},
      primaryClass={math.RA},
      url={https://arxiv.org/abs/1712.04431},
      note={preprint}
}