Branches on division algebras

TL;DR

This paper characterizes the maximal orders in 2x2 matrix algebras over non-commutative local division algebras, focusing on specific suborders like rings of integers and semisimple subalgebras.

Contribution

It provides a detailed description of maximal orders containing particular suborders in matrix algebras over division algebras, extending understanding of their structure.

Findings

Explicit descriptions of maximal orders for certain suborders

Identification of important families of suborders

Enhanced understanding of algebraic structures in non-commutative settings

Abstract

We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra B containing a given suborder, for certain important families of such suborders, including rings of integers of division subalgebras of B or most maximal semisimple commutative subalgebras.