Non-associative Categories of Octonionic Bimodules

Abstract

Category is put to work in the non-associative realm in the article. We focus on a typical example of non-associative category. Its objects are octonionic bimodules, morphisms are octonionic para-linear maps, and compositions are non-associative in general. The octonionic para-linear map is the main object of octonionic Hilbert theory because of the octonionic Riesz representation theorem. An octonionic para-linear map f is in general not octonionic linear since it subjects to the rule

\begin{array}{c}\hfill \text{Re}(f(px)-pf(x))=0.\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \text {Re}\,\big (f(px)-pf(x)\big ) =0. \end{aligned}$$\end{document}

The composition should be modified as

\begin{array}{c}\hfill f⊚g(x):=f(g(x))-\sum _{j=1}^7{e}_j\text{Re}(f(g(e_{i}x))-f(e_{i}g(x)))\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} f\circledcirc g(x):=f(g(x))-\sum _{j=1}^7 e_j\text {Re}\,\Big (f\big (g(e_ix)\big )-f\big (e_ig(x)\big )\Big ) \end{aligned}$$\end{document}

so that it preserves the octonionic para-linearity. In this non-associative category, we introduce the Hom and Tensor functors which constitute an adjoint pair. We establish the Yoneda lemma in terms of the new notion of weak functor. To define the exactness in a non-associative category, we introduce the notion of the enveloping category via a universal property. This allows us to establish the exactness of the Hom functor and Tensor functor.

Keywords

Octonions / Category / Regular composition / Para-linearity / Adjoint functor theorem

Funding

National Natural Science Foundation of China(12171448)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature