The author, [in J. Pure Appl. Algebra 163, No. 2, 193-207 (2001; Zbl 0988.16026)], classified monogenic Hopf algebras (Hopf algebras which are generated as algebras by one element), which are commutative and cocommutative (Abelian) and are local and have a local dual (local-local) over a finite field \(k\) or the Witt ring \(W(k)\) of \(k\).
In this paper, the author continues this theme, and describes the ring of Hopf algebra endomorphisms of a monogenic Abelian local-local Hopf algebra \(H\) over a finite or algebraically closed field of positive characteristic. As in the paper classifying monogenic Hopf algebras mentioned above, key to this study is the fact that \(\text{End}(H)\) is isomorphic to \(\text{End}(M)\) where \(M\) is the Dieudonné module corresponding to the group scheme represented by \(H\). Here \(M\) is a cyclic \(E\)-module of the form \(M=E/E(F^n,\alpha F^r-V)\) where \(E\) is the noncommutative polynomial ring \(W[F,V]\) over \(W=W(k)\). There is a one-to-one map from \(\text{End}(M)\) to \(P_n\), the noncommutative truncated polynomial ring \(k[F]/(F^n)\), and the author thus can calculate within this polynomial ring. The author goes on to study \(\text{End}(H)\) for \(H\) a Hopf algebra over \(W(k)\). Here it is key to note the one-to-one correspondence between isomorphism classes of finite Honda systems (these are pairs \((M,L)\) with \(M\) a Dieudonné module and \(L\) a \(W(k)\) submodule of \(M\)) and isomorphism classes of monogenic \(W(k)\)-Hopf algebras. Then \(\text{End}(H)\cong\text{End}(M,L)\); here a general classification of \(\text{End}(H)\) is not obtained but several specific cases are covered. The final section of the paper discusses possible (or impossible) generalizations of this work.