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:

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