Let \(k\) be an algebraically closed field of characteristic zero. A polyhedral group \(G\) is a finite subgroup of \(\text{PSL}_2(k)\) which is isomorphic to one of the following: (1) A finite cyclic group; (2) the dihedral group \(D_n\) of order \(2n\), \(n\) at least 2; (3) the group of symmetries of the regular tetrahedon, of order 12; (4) the group of symmetries of the regular octahedron, of order 24; or (5) the group of symmetries of the regular icosahedron, of order 60. For such a \(G\), there is a subgroup \(G'\) of \(\text{SL}_2(k)\) such that \(G'/\{1,-1\}=G\). \(G'\) is called a binary polyhedral group.
Here are the two main theorems of the paper. Theorem 1.1: Let \(H\) be a cosemisimple finite-dimensional Hopf algebra over \(k\) which contains a simple subcoalgebra \(C\) of dimension 4. Then the subalgebra \(B=k[CS(C)]\) (\(S\) is the antipode of \(H\)) is a commutative Hopf subalgebra of \(H\) isomorphic to \(k^G\), where \(G\) is a noncyclic finite subgroup of \(\text{PSL}_2(k)\) of even order. Furthermore, let \(\chi\) be the irreducible character contained in \(C\), and let \(K[\chi]\) be the stabilizer of \(\chi\) in \(H\) with respect to left multiplication in \(H\). Then \(|K[\chi]|\) divides 4, and (1) If \(|K[\chi]|=4\), then \(B\) is isomorphic to \(k^{\mathbb Z_2\times\mathbb Z_2}\); (2) If \(|K[\chi]|=2\), then \(B\) is isomorphic to \(k^{D_n}\), \(n\) at least 3; (3) If \(|K[X\chi]|=1\), then \(B\) is isomorphic to either \(k^{A_4}\), \(k^{S_4}\) or \(k^{A_5}\). – Theorem 1.1 strengthens the description of \(H\) by W. D. Nichols and M. B. Richmond [J. Pure Appl. Algebra 106, No. 3, 297-306 (1996; Zbl 0848.16034)].
Theorem 1.2: Let \(H\) be a semisimple Hopf algebra over \(k\). Assume that \(H\) has an irreducible comodule \(V\) of dimension 2 (this is equivalent to \(H\) having a simple subcoalgebra of dimension 4) such that \(V\) is faithful and self-dual. Let \(i(V)=1\) or \(-1\) denote the Frobenius-Schur indicator of \(V\). Then (1) If \(i(V)=-1\), then \(H\) is commutative and is isomorphic to \(k^{G'}\), where \(G'\) is a nonabelian binary polyhedral group. (2) If \(i(V)=1\), then either \(H\) is commutative and isomorphic to \(k^{D_n}\), \(n\) at least 3, or \(H\) is isomorphic to a deformation of a binary polyhedral group which is either generalized quaternion or cyclic. – In this discussion, \(H\) is a finite-dimensional quotient Hopf algebra of the quantum group \(SL_q(2)\) with \(q=-1\).
The proof of Theorem 1.2 uses the classification of Hopf algebras associated to nondegenerate bilinear forms in the 2 by 2 case. Theorem 1.2 leads to an equivalence of the fusion categories \(\text{Rep\,}H\) and \((\text{Vec}^G)^{\mathbb Z_2}\), where \(\text{Vec}^G\) is the fusion category of \(G\)-graded vector spaces, and the \(\mathbb Z_2\) indicates the equivariantization of \(\text{Vec}^G\) with respect to an appropriate action of \(\mathbb Z_2\). [Note: For a group \(G\), \(k^G\) refers to the dual of the group algebra \(k[G]\).]