The double sign of a real division algebra of finite dimension greater than one

TL;DR

This paper introduces the concept of double sign as an invariant for finite-dimensional real division algebras, revealing a four-block decomposition of their category and exploring structural relationships.

Contribution

It defines the double sign invariant and analyzes the categorical decomposition of real division algebras based on this invariant.

Findings

The double sign is an invariant for real division algebras.

The category of such algebras decomposes into four blocks.

Structural relationships between blocks are characterized.

Abstract

For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by a non-zero element are shown to form an invariant of A, called its double sign. The double sign causes the category of all real division algebras of a fixed dimension n>1 to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of the category of all finite-dimensional real division algebras.