{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86772"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86772","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On Hopf Algebra Type and Rational Calculus Decompositions","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Bauer, Kristine Baxter"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["McCarthy, Randy"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:26Z","date_published":"2015-09-28T15:19:26Z","updated_at":"2026-07-22T22:26:27Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3017019"],"render_values":[{"text":"(MiAaPQ)AAI3017019","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86772","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["McCarthy, Randy"]},{"key":"dc:creator","label":"Author","values":["Bauer, Kristine Baxter"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:26Z","10000-01-01","2001"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86772","(MiAaPQ)AAI3017019"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2015-09-28T15:19:26Z (GMT). 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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.","Made available in DSpace on 2015-09-28T15:19:26Z (GMT). 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