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Decomposition of Topological Azumaya Algebras with Orthogonal Involution

Niny Arcila-Maya

TL;DR

This work investigates when a topological Azumaya algebra of degree $mn$ over a CW complex $X$ with an orthogonal involution can be expressed as a tensor product of degree $m$ and $n$ algebras carrying orthogonal involutions, under the hypotheses $\gcd(m,n)=1$ and $\dim(X)\le\min\{m,n\}$. Employing homotopy-theoretic tools and stabilization in the complex orthogonal group, the paper constructs lifts through classifying spaces $\mathrm{BPO}$ and $\BSO$, and analyzes tensor-product operations on homotopy groups to derive conditions for decomposition. The main result, Theorem $\text{['mainO']}$, shows that in the mixed-parity case ($m$ even, $n$ odd) the algebra $\mathcal{A}$ decomposes as $\mathcal{A}_m\otimes\mathcal{A}_n$ with $\mathcal{A}_n$ Brauer-trivial, while an analogous decomposition holds for orthogonal vector bundles in the odd-odd case. The paper also discusses obstructions to decomposition outside the stable range, and extends the tensor-product decomposition to orthogonal bundles, highlighting the interplay between Azumaya theory, involutions, and topological $K$-theory in low dimensions.

Abstract

Let $\mathcal{A}$ be a topological Azumaya algebra of degree $mn$ with an orthogonal involution over a CW complex $X$ of dimension less than or equal to $\min\{m,n\}$. We give conditions for the positive integers $m$ and $n$ so that $\mathcal{A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $m$ and $n$ with orthogonal involutions.

Decomposition of Topological Azumaya Algebras with Orthogonal Involution

TL;DR

This work investigates when a topological Azumaya algebra of degree over a CW complex with an orthogonal involution can be expressed as a tensor product of degree and algebras carrying orthogonal involutions, under the hypotheses and . Employing homotopy-theoretic tools and stabilization in the complex orthogonal group, the paper constructs lifts through classifying spaces and , and analyzes tensor-product operations on homotopy groups to derive conditions for decomposition. The main result, Theorem , shows that in the mixed-parity case ( even, odd) the algebra decomposes as with Brauer-trivial, while an analogous decomposition holds for orthogonal vector bundles in the odd-odd case. The paper also discusses obstructions to decomposition outside the stable range, and extends the tensor-product decomposition to orthogonal bundles, highlighting the interplay between Azumaya theory, involutions, and topological -theory in low dimensions.

Abstract

Let be a topological Azumaya algebra of degree with an orthogonal involution over a CW complex of dimension less than or equal to . We give conditions for the positive integers and so that can be decomposed as the tensor product of topological Azumaya algebras of degrees and with orthogonal involutions.
Paper Structure (17 sections, 26 theorems, 40 equations, 4 tables)

This paper contains 17 sections, 26 theorems, 40 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notation
  4. Preliminaries
  5. Involutions on matrix algebras
  6. Non-degenerate bilinear forms
  7. Stabilization of operations on the complex orthogonal group
  8. First unstable homotopy group of On
  9. Stabilization
  10. Operations on homotopy groups
  11. Tensor product on the quotient
  12. Case $m$ and $n$ odd
  13. Case $m$ even and $n$ odd
  14. Proof of Theorem \ref{['mainO']}
  15. A d-connected map where d is the minimum of two positive integers of opposite parity
  16. ...and 2 more sections

Key Result

Theorem 1.5

Let $m$ and $n$ be relatively prime positive integers such that $m$ is even, and $n$ is odd. Let $X$ be a CW complex such that $\dim(X)\leq d$ where $d\coloneqq\min\{m,n\}$. If $\mathcal{A}$ is a topological Azumaya algebra of degree $mn$ over $X$ with an orthogonal involution, then there exist topo

Theorems & Definitions (46)

  • Example 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Remark 2.2
  • Lemma 3.2
  • proof
  • ...and 36 more