Shira Gilat | Searcharxiv

arXiv2023

The algebra of supernatural matrices

The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras, which studied in a previous paper of the authors (\cite{Local}). It is also a stand-alone object admits a rich study and various connections to other fields. The goal of this paper is to expose some new information about supernatural matrices, mainly in terms of the "inner" ways to identify such algebras, and their appearance as minimal solutions to equations of the form $M_n(A)\cong A$. 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.

math.RA↗

arXiv2017

Linkage of Quadratic Pfister Forms

We study the necessary conditions for sets of quadratic $n$-fold Pfister forms to have a common $(n-1)$-fold Pfister factor. For any set $S$ of $n$-fold Pfister forms generating a subgroup of $I_q^n F/I_q^{n+1} F$ of order $2^s$ in which every element has an $n$-fold Pfister representative, we associate an invariant in $I_q^{n+1} F$ which lives inside $I_q^{n+s-1} F$ when the forms in $S$ have a common $(n-1)$-fold Pfister factor. We study the properties of this invariant and compute it explicitly in a few interesting cases.

math.RA↗