On Hopf Algebra Type and Rational Calculus Decompositions
The second part of my thesis, which is joint work with Randy McCarthy, uses Goodwillie calculus to extend this result to a much larger class of functors. A Hopf algebra A is both an algebra with a multiplication map m:A⊗A→ A and a coalgebra with a comultiplication map D: A→A⊗A which must behave well with respect to each other. Mimicking this definition, we say that an object X of any category which has coproducts, ∨ , is of Hopf algebra type if there is a map 1:X→X∨X which acts like the comultiplication with respect to the fold map, which acts like the multiplication. Randy McCarthy and I have been able to show that rationally, the Goodwillie calculus tower of homotopy functors evaluated on objects of Hopf algebra type split, providing a decomposition. Furthermore, this decomposition generalizes the decomposition of higher Hochschild homology of Part I. Other examples include the cohomology of loop spaces and the Poincare-Birkhoff Witt theorem for Lie algebras over fields of characteristic zero.
University of Illinois - Urbana-Champaign
Networked Digital Library of Theses and Dissertations
University of Illinois at Urbana-Champaign