Adjacency preserving maps on classical spaces of tensors / Lau Jin Ting

Lau , Jin Ting (2024) Adjacency preserving maps on classical spaces of tensors / Lau Jin Ting. PhD thesis, Universiti Malaya.

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Abstract

We define tensor rank, symmetric rank and exterior rank of tensor, symmetric tensors and multivectors, respectively. A pair of tensors, symmetric tensors or multivectors is adjacent if the rank of their difference is one. A map on tensor spaces, symmetric spaces or exterior spaces is strong adjacency preserving provided it preserves adjacent pairs in both directions. In this thesis, we characterize strong adjacency preserving maps on tensor spaces of order at least three, and classify surjective strong adjacency preserving maps on symmetric spaces and exterior spaces. Our strategy is to extract properties of rank one tensors, symmetric rank one tensors and exterior rank one multivectors, and reduce surjective strong adjacency preserving maps to certain maps on affine spaces or projective spaces. Additive maps on exterior spaces that preserve decomposable multivectors are closely related to strong adjacency preserving maps. We characterize this class of additive maps by reducing them to linear maps through the field extensions approach, and deduce the structure of such additive maps from the linear maps.

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