Introduction to Hopf algebras in renormalization and noncommutative geometry

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We review the appearance of Hopf algebras in the renormalization of quantum field theories and in the study of diffeomorphisms of the frame bundle important for index computations in noncommutative geometry.
12 pages, 1 eps figure, LaTeX 2e; based on talk at workshop `The Standard Model of Elementary Particle Physics from a Mathematical-Geometrical Viewpoint', Hesselberg, March 1999; to appear in proceedings (written in May 1999, recent developments not covered, hopefully still useful)

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