Semisimple Hopf algebras
The authors study the structure of finite dimensional semisimple Hopf algebras over a field \(K\), using the trace formula, the Nichols-Zoeller theorem [\textit{W. D. Nichols} and \textit{M. B. Zoeller}, J. Pure Appl. Algebra 56, 51-57 (1989; Zbl 0659.16006)] and the authors' results in other papers. Semisimple Hopf algebras are unimodular, but the converse is not always true. In Section 2, the authors show that if \(A\) is an odd- dimensional Hopf algebra over a field \(K\) and \(\text{char }K = 0\) or 2 or \(p > \dim A\), then \(A\) and \(A^*\) are semisimple iff \(A\) and \(A^*\) are unimodular. Also, it is shown that if \(A\) and \(A^*\) are unimodular and \((\dim A) 1 \neq 0\) then \(A\) and \(A^*\) are semisimple when \(A\) has an involutory sub-Hopf algebra \(B\) such that \(\text{rank}_ B A \leq 3\). In Section 3 the authors study adjoint module actions on a semisimple Hopf algebra \(A\) over an algebraically closed field \(K\) and show that if the minimal ideals of \(A\) have dimension 1 or \(n^ 2\) for a single \(n > 1\), then \(n\) divides the number of one-dimensional ideals of \(A\) (and hence \(n\) divides \(\dim A\)). In Section 4, the results in Section 2 and 3 are applied to study \(n\)- dimensional semisimple Hopf algebras when \(n \leq 20\). The authors show that if \(n \leq 19\) and is odd then \(A\) is commutative and cocommutative (hence \(A^*\) was the form of a group algebra). In particular, all 5- dimensional Hopf algebras are commutative and cocommutative. If \(n \leq 20\) and is even, the semisimple Hopf algebra \(A\) is involutory when \((\dim A)1 \neq 0\). The techniques developed in this paper are quite useful in determining potential candidates for noninvolutory semisimple Hopf algebras \(A\) over \(K\). In Section 5 a systematic computation is described which enumerates possible algebra structures for such Hopf algebras of dimension \(\leq 50\). In Section 6 a result is given which further limits possible algebra structures for noninvolutory semisimple Hopf algebras whose duals are unimodular. \textit{R. Williams} [Finite dimensional Hopf algebras (Thesis, Florida State Univ. 1988)] has classified Hopf algebras of \(\dim \leq 11\).
finite dimensional semisimple Hopf algebras
scientific article; zbMATH DE number 734556