Ultrahomogeneous tensor spaces

2022/07/20 by Harman, Nate, Snowden, Andrew · 1 citation
#FOS: Mathematics #Logic (math.LO) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2207.09626

Abstract

A cubic space is a vector space equipped with a symmetric trilinear form. Using categorical Fraïssé 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 ω-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.

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• A remark on automorphisms of tensor spaces

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