Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 053, 42 pages      arXiv:2104.08895      https://doi.org/10.3842/SIGMA.2022.053
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

Ralph M. Kaufmann ab and Yang Mo a
a) Department of Mathematics, Purdue University, West Lafayette, IN, USA
b) Department of Physics and Astronomy, Purdue University, West Lafayette, IN, USA

Received April 18, 2021, in final form June 29, 2022; Published online July 11, 2022

Abstract
We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics. In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the appearance of Brown style coactions. Using previous results, we can interpret all the relevant coalgebras as stemming from a categorical construction, tie the bialgebra structures to Feynman categories, and apply the developed theory in this setting.

Key words: Feynman category; bialgebra; Hopf algebra; antipodes; renomalization; characters; combinatorial coalgebra; graphs; trees; Rota-Baxter; colored structures.

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