Smash product

Summary

The smash product $A \# H$ is an algebra built from a Hopf algebra $H$ and an algebra $A$. The Hopf algebra $H$ acts on $A$ compatibly with the multiplication of $A$. It contains $A$ and $H$ as subalgebras, and the action of $H$ on $A$ becomes part of the multiplication. For the group algebra of a group it is the skew group ring. For the enveloping algebra of a Lie algebra acting by derivations it is a differential operator ring. So it treats group actions and actions by derivations in one framework.