Products, coproducts and singular value decomposition

TL;DR

This paper explores how singular value decomposition applied to products and coproducts in monoidal categories reveals invariant data, with implications for noncommutative geometry and algebraic structures like Grassmann and Clifford algebras.

Contribution

It demonstrates how twist maps influence singular value data in products and coproducts, providing insights into noncommutative geometries and algebraic structures.

Findings

Twist maps reduce degeneracies in singular value data.

Coproducts for positive eigenvalues produce corresponding eigenvectors in tensor products.

Insights into noncommutative behavior of algebra of functions.

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, 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, 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.