Cocommutative Hopf algebra

Symbol-free definition
A Hopf algebra is said to be cocommutative if the comultiplication operation therein is cocommutative.

Definition with symbols
A Hopf algebra $$H$$ with a comultiplication $$\Delta$$ is said to be comultiplication if $$\Delta$$ is cocommutative, or in other words, if:

$$\Delta = \tau \circ \Delta$$

where $$\tau$$ is the unique linear map that sends each pure tensor $$x \otimes y$$ to $$y \otimes x$$.

Related notions

 * Commutative Hopf algebra
 * Quasitriangular Hopf algebra