Averaging operators on groups and Hopf algebras
Huhu Zhang, Xing Gao
TL;DR
This work introduces averaging operators on groups and Hopf algebras, motivated by a Koszul duality perspective with weight-zero Rota-Baxter operators and by tangent-space considerations to Lie theory. It defines averaging groups and averaging Hopf algebras, shows that an averaging group naturally induces a disemigroup and a rack, and proves that differentiating a smooth averaging operator on a Lie group yields an averaging Lie algebra. It further establishes a correspondence between averaging operators on a group and averaging operators on its group Hopf algebra, and provides an explicit, fully combinatorial construction of the free averaging group on a set. Together, these results connect group-theoretic, Lie-algebraic, and Hopf-algebraic operator theories and furnish concrete tools for manipulating averaging structures within algebraic combinatorics and related areas of mathematical physics.
Abstract
Rota-Baxter operators on groups were studied quite recently. Motivated mainly by the fact that weight zero Rota-Baxter operators and averaging operators are Koszul dual to each other, we propose the concepts of averaging group and averaging Hopf algebra, and study relationships among them and the existing averaging Lie algebras. We also show that an averaging group induces a disemigroup and a rack, respectively. As the free object is one of the most significant objects in a category, we also construct explicitly the free averaging group on a set.