# Group algebra as a Hopf algebra

## Definition

Let be a group and a field. The **group algebra** over , when talked of as a Hopf algebra, is the following:

- The unital associative algebra part is the same as for the usual group algebra: We consider a vector space whose basis is indexed by elements of the group, and define multiplication of these basis elements by multiplication in the group.
- The comultiplication is defined by linearly extending the map:

for every

In other words:

- The counit is defined by linearly extending the map:

for all

In other words:

- The antipode map is defined by linearly extending the map:

In other words: