Monoidal structure on Rota-Baxter algebras via augmentation, and free cocycle bialgebras

Abstract

A tensor product is known for commutative Rota--Baxter algebras through a universal construction, but its underlying algebra need not be the ordinary tensor product $A\otimes B$ and can be substantially larger. We show that, after passing to augmented Rota--Baxter algebras, the ordinary tensor product itself carries a natural Rota--Baxter operator which makes the category monoidal with the usual associator and unit constraints. The resulting structure is intrinsically nonsymmetric: over a field of characteristic zero, this operator and its flip-conjugated counterpart are the only possibilities among linear combinations of five natural terms, while over any field the full category admits no braiding. Its comonoid objects are precisely Rota--Baxter cocycle bialgebras. Moreover, the free Rota--Baxter adjunction yields a bimonad and free cocycle bialgebras on ordinary bialgebras, with compatible Hopf structures for connected filtered inputs, in both the associative and commutative settings.