nLab infinitesimal bialgebra

An infinitesimal bialgebra (abbreviated $\epsilon$-bialgebra) over a commutative unital ring $k$ is a $k$-module $A$ which is equipped with the structures of an associative unital $k$-algebra $(A,m,\eta)$, coassociative counital $k$-coalgebra $(A,\Delta,\epsilon)$ and such that the following compatibility condition holds: the comultiplication $\Delta:A\to A\otimes A$ is a derivation, where $A\otimes A$ is equipped with the standard algebra and standard (external) bimodule structure, i.e. $\Delta(a b) = \Delta(a)(1\otimes b) + (a\otimes 1)\Delta(b)$.

• Marcelo Aguiar, On the associative analogues of Lie bialgebras, pdf, Infinitesimal Hopf algebras, pdf