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GR: Gravitation und Relativitätstheorie

GR 7: Grundlegende Probleme und allgemeiner Formalismus

GR 7.3: Fachvortrag

Donnerstag, 18. März 2004, 09:40–10:00, H20

Products, Coproducts, and Singular Value Decomposition — •Bertfried Fauser — MPI Leipzig, Inselstrasse 22, D-04103 Leipzig

Points in ordinary space are usually described via vectors and/or coordinate tuples. Considering the function algebra –or for sake of simplicity polynomial algebra– on such point spaces leads to the point wise product of these functions. Coordinates are assumed to be commutative. Recasting these facts in terms of Hopf algebras one obtains group like coproducts, allowing to generalize the concept of a point using non group like coproducts. A product map is a map m(V⊗ V)→ V. In categorial terms we can regard it as map m:W→ V with W=V⊗ V, hence we obtain rectangular matrix representations, similarly for coproducts. Usually one obtains admissible coproducts from categorially dualizing product structures. For the case of Grassmann and Clifford products we develop a spectral theory of multiplication-comultiplication pairs. Certain interesting consequences can be obtained an a spectral description of the products and coproducts is derived. Such a spectral basis might help simplifying calculations and to gain geometrical insight into the mere structure of the underlying point space.