Products, coproducts and singular value decomposition

Bertfried Fauser

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Submission date: 28. Feb. 2004
Pages: 20
published in: International journal of theoretical physics, 45 (2006) 9, p. 1731-1755 
DOI number (of the published article): 10.1007/s10773-006-9111-6
Bibtex
MSC-Numbers: 15A18, 16W30, 15A66
Keywords and phrases: singular value decomposition, hopf algebra, clifford and grassmann algebra
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Abstract:
Products and coproducts may be recognized as morphisms in a monoidal tensor category of vector spaces. To gain invariant data of these morphisms, we can use singular value decomposition which attaches singular values, \textit{ie} generalized eigenvalues, to these maps. We show, for the case of Grassmann and Clifford products, that twist maps significantly alter these data reducing degeneracies. Since non group like coproducts give rise to non classical behavior of the algebra of functions, \textit{ie} make them noncommutative, we hope to be able to learn more about such geometries. Remarkably the coproduct for positive singular values of eigenvectors in $A$ yields directly corresponding eigenvectors in $A\otimes A$.