The algebra of supernatural matrices

Tamar Bar-On; Shira Gilat; Eliyahu Matzri; Uzi Vishne

doi:10.1142/S0219498826501550

The algebra of supernatural matrices

Tamar Bar-On
, Shira Gilat
, Eliyahu Matzri
, Uzi Vishne

Research output: Contribution to journal › Article › peer-review

Abstract

The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras, which we developed in a previous paper [T. Bar-On, Sh. Gilat, E. Matzri and U. Vishne, Locally finite central simple algebras, Algebras Represent. Theory 26(2) (2023) 553–607]. This algebra has appeared under various names before, and deserves further study. Supernatural matrices are a minimal solution to the equation of unital algebras Mn(X) ^∼= X, which we compare to several similar conditions involving cancellation of matrices. Viewing a natural representation of this algebra, we show that supernatural matrices generalize both McCrimmon’s deep matrices algebra and m-petal Leavitt path algebra. We also study their simple representations.

Original language	English
Article number	2650155
Journal	Journal of Algebra and its Applications
DOIs	https://doi.org/10.1142/S0219498826501550
State	Accepted/In press - 1 Jan 2025
Externally published	Yes

Keywords

Leavitt path algebras
Supernatural matrices
central simple algebras
deep matrices
direct limit
infinite Brauer monoid
locally finite-dimensional algebras
matrix cancellation

ASJC Scopus subject areas

Algebra and Number Theory
Applied Mathematics

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10.1142/S0219498826501550

Cite this

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title = "The algebra of supernatural matrices",

abstract = "The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras, which we developed in a previous paper [T. Bar-On, Sh. Gilat, E. Matzri and U. Vishne, Locally finite central simple algebras, Algebras Represent. Theory 26(2) (2023) 553–607]. This algebra has appeared under various names before, and deserves further study. Supernatural matrices are a minimal solution to the equation of unital algebras Mn(X) ∼= X, which we compare to several similar conditions involving cancellation of matrices. Viewing a natural representation of this algebra, we show that supernatural matrices generalize both McCrimmon{\textquoteright}s deep matrices algebra and m-petal Leavitt path algebra. We also study their simple representations.",

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AU - Gilat, Shira

AU - Matzri, Eliyahu

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KW - Supernatural matrices

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KW - direct limit

KW - infinite Brauer monoid

KW - locally finite-dimensional algebras

KW - matrix cancellation

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The algebra of supernatural matrices

Abstract

Keywords

ASJC Scopus subject areas

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