Rota-Baxter Algebras, Renormalization on Kausz Compactifications and Replicating of Binary Operads

Abstract

This thesis is divided into two parts: In the first part, we consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole subtraction. We apply this formalism to the unrenormalized momentum space Feynman amplitudes, viewed as (divergent) integrals in the complement of the determinant hypersurface. We lift the integral to the Kausz compactification of the general linear group, whose boundary divisor is normal crossings. We show that the Kausz compactification is a Tate motive and the boundary divisor is a mixed Tate configuration. The regularization of the integrals that we obtain differs from the usual renormalization of physical Feynman amplitudes, and in particular it gives mixed Tate periods in cases that have non-mixed Tate contributions in the usual form. This part is based on joint work with Matilde Marcolli (see (80)). In the second part, we consider the notions of the replicators, including the duplicator and triplicator, of a binary operad. We show that taking replicators is in Koszul dual to taking successors in (9) for binary quadratic operads and is equivalent to taking the white product with certain operads such as Perm. We also relate the replicators to the actions of average operators. After the completion of this work (in 2012; see (85)), we realized that the closely related notions di-Var-algebra and tri-Var-algebra have been introduced independently in (48) (in 2011; see also (63; 64)) by Kolesnikov and his coauthors. In fact their notions also apply to not necessarily binary operads (64). In this regard, the second part of this thesis provides an alternative and more detailed treatment of these notations for binary operads. This part is based on joint work with Chengming Bai, Li Guo, and Jun Pei (see (85)).