Definition. the Group Algebra and its universal extension [fgap-001H]
Definition. the Group Algebra and its universal extension [fgap-001H]
Let \(k\) be a commutative ring and \(G\) a group. The Group Algebra \(k[G]\) is the free \(k\)-module with basis \([g]\) indexed by \(g\in G\), equipped with \[ [g][h]=[gh],\qquad 1=[1_G],\qquad (\sum _g a_g[g])(\sum _h b_h[h])=\sum _{g,h}a_gb_h[gh]. \] The scalar embedding sends \(a\) to \(a[1_G]\). Thus each group element is a unit of \(k[G]\), with inverse \([g^{-1}]\).
If \(A\) is a \(k\)-algebra and \(u:G\to A^\times \) is a group homomorphism, there is a unique \(k\)-algebra homomorphism \[ \widetilde u:k[G]\longrightarrow A, \qquad \widetilde u\left (\sum _g a_g[g]\right )=\sum _g a_g u(g). \] This universal property extends representations and concrete group maps from basis elements to the whole Group Algebra. See [sengupta2010representations, secs. 3.1--3.2, pp. 39--41].