Self-dual Hopf algebras of dimension \(p^ 3\) obtained by extension
The author continues the recent progress in classifying Hopf algebras of dimensions \(p\) and \(p^2\), \(p\) a prime, to the case of dimension \(p^3\). Let \(C_p\) be the cyclic group of order \(p\), \(p\) an odd prime. The author constructs a class of extensions \(A\) of \(k[C_p \times C_p]^*\) by \(kC_p\), where \(*\) is the \(k\)-linear dual, \(k\) a field. \(A = A_{\xi, g}\) depends on a \(p\)-th root of unity \(\zeta\) in \(k\), and certain group-like elements \(g\) in \(k[C_p \times C_p]^*\). \(A_{\xi,g}\) is self-dual if \(\xi \neq 1\). The main result is that if \(k\) has characteristic zero, then, up to isomorphism, there are \(p + 1\) semisimple Hopf algebras of dimension \(p^3\) which are neither commutative nor cocommutative, each isomorphic to some \(A_{\xi,g}\) with \(\xi \neq 1\). The case \(p = 2\) had been handled previously by the author, there being only one semisimple Hopf algebra (characteristic \(k \neq 2)\) of dimension 8 which is neither commutative nor cocommutative.
semisimple Hopf algebras of dimension \(p^ 3\)
scientific article; zbMATH DE number 840730