Decomposition of Topological Azumaya Algebras

TL;DR

This paper investigates conditions under which topological Azumaya algebras can be decomposed into tensor products of lower-degree algebras, revealing limitations based on algebra degree and space dimension.

Contribution

It provides new criteria for decomposing topological Azumaya algebras and demonstrates cases where such decompositions are impossible.

Findings

Decomposition criteria depend on algebra degree and space topology.

High-dimensional spaces can prevent algebra decomposition.

Certain degrees of Azumaya algebras cannot be decomposed in specified conditions.

Abstract

Let $\mathcal{A}$ be a topological Azumaya algebra of degree $mn$ over a CW complex $X$. We give conditions for the positive integers $m$ and $n$, and the space $X$ so that $\mathcal{A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $m$ and $n$. Then we prove that if $m<n$ and the dimension of $X$ is higher than $2m+1$, $\mathcal{A}$ may not have such decomposition.