The TIZ-correspondence adjusted for symmetry

Abstract

The TIZ-correspondence ([1], Theorem B) is a ternary Galois correspondence between generalized tensor products, polynomial ideals, and affine schemes of tensor operators. We study the TIZ-correspondence under the presence of symmetry. We provide evidence that this correspondence does not have an internal characterization of symmetry, and we propose three definitions of a generalized symmetric tensor product. For each version, we prove a variant of the TIZ-correspondence in this setting. For the last and most general version, we prove that Lie algebras naturally coordinatize these generalized symmetric tensor products. We prove that every symmetric multilinear map t has a universally smallest generalized tensor product space containing t. We additionally survey the main results in [1]. We give proofs of theorems A-D, demonstrate each theorem with examples, and we provide explicit computations of many of the objects involved.