Ultrahomogeneous tensor spaces

TL;DR

This paper constructs a unique, universal ultrahomogeneous cubic tensor space of countable dimension, analyzes its automorphism group, and explores its model-theoretic properties, revealing deep structural and symmetry characteristics.

Contribution

It introduces a universal ultrahomogeneous tensor space using categorical Fraïssé theory and characterizes its automorphism group and model-theoretic features, extending to general tensor spaces.

Findings

Existence of a unique universal ultrahomogeneous cubic space

Automorphism group is a linear-oligomorphic group

The space is ω-categorical with quantifier elimination

Abstract

A cubic space is a vector space equipped with a symmetric trilinear form. Using categorical Fra\"iss\'e theory, we show that there is a universal ultrahomogeneous cubic space $V$ of countable infinite dimension, which is unique up to isomorphism. The automorphism group $G$ of $V$ is quite large and, in some respects, similar to the infinite orthogonal group. We show that $G$ is a linear-oligomorphic group (a class of groups we introduce), and we determine the algebraic representation theory of $G$. We also establish some model-theoretic results about $V$: it is $\omega$-categorical (in a modified sense), and has quantifier elimination (for vectors). Our results are not specific to cubic spaces, and hold for a very general class of tensor spaces; we view these spaces as linear analogs of the relational structures studied in model theory.