Skew derivations of finite-dimensional algebras and actions of the double of the Taft Hopf algebra
Let \(H\) be the \(n^2\)-dimensional Taft Hopf algebra generated by elements \(g\) and \(x\), \(g\) group-like, \(x\) \((g,1)\)-primitive, \(g^n=1\), \(x^n=0\), \(xg=wxg\), \(w\) a primitive \(n\)-th root of \(1\) in the base field \(k\), (\(n>1\)). \(H\) is a homomorphic image of \(L\), where the relation \(x^n=0\) is omitted. By studying \(L\)-actions on certain algebras, the authors obtain information about the actions of \(H\) on the algebra \(A=k[u]\) with \(u^n\) in \(k\), i.e., skew derivations of \(A\). They also determine the actions of the Drinfeld double \(D(H)\) on \(A\), in particular those giving Yetter-Drinfeld structures. As an application, let \(u_q(\text{sl}_2)=k[E,F,K]\) be the finite-dimensional quotient of \(U_q(\text{sl}_2)\) defined by Lusztig, \(q\) a \(2n\)-th root of \(1\). The authors show that the actions of \(u_q(\text{sl}_2)\) on \(A\) are determined by the action of a Borel subalgebra, i.e., by the action of \(E\) (or \(F\)) on \(u\). They also construct smash products of \(A\) and \(u_q(\text{sl}_2)\) which are not semisimple. Thus a result of Azumaya which states that if \(G\) is a finite group with an outer action on a simple algebra \(A\), then the smash product of \(A\) and the group algebra \(kG\) is simple, does not generalize to actions of pointed Hopf algebras.
scientific article; zbMATH DE number 1822090