Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gröbner-Shirshov bases
arXiv:2002.02864 · doi:10.2140/pjm.2022.317.441
Abstract
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted forests with multiple decoration sets to construct free Hopf algebras with multiple Hochschild 1-cocycle conditions. Applying the universal property of the underlying operated algebras and the method of Gröbner-Shirshov bases, we then construct free objects in the category of matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebras to allow multiple Rota-Baxter operators. Finally the free matching Rota-Baxter algebras are equipped with a cocycle Hopf algebra structure.
27 pages
References in corpus (9)
- Hopf algebras, from basics to applications to renormalization
- Differential Type Operators and Gröbner-Shirshov Bases
- Operads of compatible structures and weighted partitions
- Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles
- Hopf algebras of planar binary trees: an operated algebra approach
- Braided dendriform and tridendriform algebras and braided Hopf algebras of planar trees
- Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras
- Locality and renormalisation: universal properties and integrals on trees
- Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebras