---
tag: 2ZJ2G308QV
kind: concept
name: Smash product
msc: [16S40, 16T05]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Hopf_smash_product
---

## Summary

The smash product $A \# H$ is an algebra built from a [[Hopf algebra|7XFYP0N84H]] $H$ 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|PCKK4SFZEK]].
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.

## Explanations

### L1 human

A group can act on an algebra by symmetries, and a Lie algebra can act on it by derivations.
A Hopf algebra is a common generalisation of both, and it can also act on an algebra.
The smash product puts the algebra and the Hopf algebra side by side, as $A \otimes H$.
When an element $h$ of $H$ moves past an element $b$ of $A$, it acts on $b$.
This works in the same way that a group element moves past a coefficient in a skew group ring.