2003/04/20 by Daniel Henry Gottlieb, Gottlieb, Daniel Henry
Mathematics · Physics and Astronomy · #15A63 #17B90 #57R45 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Condensed Matter (cond-mat) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Spectral Theory (math.SP) #cond-mat #gr-qc #math.AT #math.SP #msc:15A63 #msc:17B90 #msc:57R45
paper · pdf · doi:10.48550/arxiv.math/0304281
22 pages, also found on http://math.purdue.edu/~gottlieb
arxiv created 2003/04/20 · openalex publication_date 2003/04/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a parameterized space of square matrices, the associated set of eigenvectors forms some kind of a structure over the parameter space. When is that structure a vector bundle? When is there a vector field of eigenvectors? We answer those questions in terms of three obstructions, using a Homotopy Theory approach. We illustrate our obstructions with five examples. One of those examples gives rise to a 4 by 4 matrix representation of the Complex Quaternions. This representation shows the relationship of the Biquaternions with low dimensional Lie groups and algebras, Electro-magnetism, and Relativity Theory. The eigenstructure of this representation is very interesting, and our choice of notation produces important mathematical expressions found in those fields and in Quantum Mechanics. In particular, we show that the Doppler shift factor is analogous to Berry's Phase.