nLab
module algebra

A left module $k$-algebra is a $k$-algebra $A$ equipped with a left Hopf action (also sometimes said module algebra action) $▹:B\otimes A\to A$ of a $k$-bialgebra $B$. Sometimes one talks about module algebras meaning a monoid with Hopf actions of a bimonoid in a more general symmetric monoidal category; a module monoid would be a better term if the category is not $k$-linear.

Related entries include smash product algebra, comodule algebra, gebra, bigebra

• S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.
• Susan Montgomery, Hopf algebras and their actions on rings, CBMS Lecture Notes 82, AMS 1993, 240p.
• Matthew Tucker-Simmons, $*$-structures on module-algebras, Ph. D. thesis, arxiv/1211.6652