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:

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