Group algebra as a Hopf algebra

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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}