proposition. the standard Hopf structure on a Group Algebra [fgap-001V]
✍️source
MonoidAlgebra.instHopfAlgebraMonoidAlgebra.counit_singleMonoidAlgebra.comul_singleMonoidAlgebra.antipode_single
L∃∀N

    Let \(k\) be a commutative ring and \(G\) a group. The Group Algebra \(k[G]\) is a Hopf algebra with structure determined on basis elements by \[ \Delta ([g])=[g]\otimes [g],\qquad \epsilon ([g])=1,\qquad S([g])=[g^{-1}]. \] The comultiplication and counit are algebra homomorphisms; the antipode extends inversion and satisfies the 2 convolution identities.

    Proof. 

      Each formula respects multiplication on basis elements. For example, \(\Delta ([gh])=[gh]\otimes [gh]=([g]\otimes [g])([h]\otimes [h])\). The counit calculation is similar. Finally \([g][g^{-1}]=[1]=[g^{-1}][g]\), which gives both antipode identities on the basis and hence by linearity on all of \(k[G]\). This group-algebra Hopf structure appears, for example, in [broue2024rings, ex. 18.3.3(1), pp. 500--501].