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  1. ; (, Advances in mathematics)
    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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