On the antipode of a cosemisimple Hopf algebra
Let H be a Hopf algebra over a field k which is cosemisimple (direct sum of simple subcoalgebras). When H is finite-dimensional over k, it is an open question whether or not the antipode s of H satisfies \(s^ 2=I\). \textit{R. G. Larson} showed [Commun. Algebra 11, 1175-1186 (1983; Zbl 0514.16005)] that if each simple subcoalgebra C of H has dimension \(\leq 8\), and if k has characteristic 0 or \(p>4 \dim H\), then \(s^ 2=I\). By extending to the algebraic closure of k, this means C is a full matrix coalgebra of degree \(\leq 2\) and that \(s^ 2\) is diagonalizable. In the paper under review, the author generalizes this result to show that if H is a cosemisimple Hopf algebra over k (possibly infinite-dimensional) with antipode s, such that \(s^ 2\) is diagonalizable and H is generated by its subcoalgebras of dimension \(\leq 8\), then s has order 1, 2 or \(\infty\). The proof involves the study of certain free Hopf algebras H(C,T) where C is a coalgebra and T a coalgebra endomorphism of C. There is a canonical map i:\(C\to H(C,T)\) such that \(iT=S^ 2i\), where S is the antipode of H(C,T), and the construction is universal. The author obtains an explicit basis for H(C,T) when C is a full matrix coalgebra, and T an endomorphism relevant to his proof of the above theorem. When k contains an element transcendental over the prime field, the author constructs an automorphism T of the full 2 by 2 matrix coalgebra C such that H(C,T) is cosemisimple, \(S^ 2\) is diagonalizable and S has infinite order. Thus the possibility of order \(\infty\) in the above theorem can in fact be realized (in a necessarily infinite-dimensional example).
scientific article; zbMATH DE number 3842069