Exercise 7, § 1

Suppose that K is a field. Let E be a bigebra satisfying the following conditions: (i) E is cocommutative; (ii) E has an inversion (Algebra, Chapter III, § 11, Exercise 4); (iii) $P(E) = \{0\}$; (iv) E is of finite rank over K. Let $E'$ denote the dual K-algebra of the cogebra E. Let G denote the set of elements $g \in E$ such that $\varepsilon(g) = 1$ and $c(g) = g \otimes g$; it is a group (cf. Exercise 4). (a) Let $g \in G$ and let $m_g$ be the kernel of the homomorphism $u \mapsto u(g)$ of $E'$ into K. Show that $m_g = m_g^2$ (reduce it to the case $g = 1$ using Exercise 6; then apply Exercise 5). (b) Suppose that K is algebraically closed. Show that the ideals $m_g$ are the only maximal ideals of $E'$; deduce that their intersection is $\{0\}$, that $E'$ is identified with the product $K^G$ and that E is identified with the bigebra $K[G]$ of the finite group G (cf. Exercise 3).