Group algebra as a Hopf algebra

Definition
Let $$G$$ be a group and $$K$$ a field. The group algebra over $$G$$, 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:

$$g \mapsto g \otimes g$$ for every $$g$$

In other words:

$$\nabla(\sum a_g g) = \sum a_g (g \otimes g)$$


 * The counit is defined by linearly extending the map:

$$g \mapsto 1$$ for all $$g$$

In other words:

$$\epsilon(\sum a_g g) = \sum a_g$$


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

$$g \mapsto g^{-1}$$

In other words:

$$S(\sum a_g g) = \sum a_g g^{-1}$$