The Grothendieck group of a Hopf algebra (Q1910766)

\textit{I. Kaplansky} conjectured [in Bialgebras (Univ. Chicago Lect. Notes 1975)] that if a finite-dimensional cosemisimple Hopf algebra \(H\) over an algebraically closed field contains a simple subcoalgebra of dimension \(n^2\), then \(n\) divides the dimension of \(H\). In this paper, the authors verify this conjecture when \(H\) contains a simple subcoalgebra of dimension 4. The main result is: if \(H\) is a cosimple Hopf algebra over an algebraically closed field and contains a simple subcoalgebra of dimension 4, then \(H\) contains either a Hopf subalgebra of dimension 2, 12 or 60, or a simple subcoalgebra of dimension \(n^2\) for each positive integer \(n\). To achieve this generality, the Grothendieck group of the category of finite-dimensional right \(H\)-comodules is defined and is studied. Some interesting properties are obtained. As an application, the main result is proved. As a corollary, if \(H\) is still finite dimensional, then it has even dimension.