Green rings of pointed rank one Hopf algebras of nilpotent type.
For a Hopf algebra \(A\) over a field \(k\), let \(F(A)\) be the free abelian group generated by isomorphism classes \([V]\) of finite-dimensional \(A\)-modules \(V\). \(F(A)\) is a ring using tensor products. The Green ring \(r(A)\) is the quotient ring of \(F(A)\) modulo the relations \([V]+[W]\) is the class of the direct sum of \(V\) and \(W\). \(r(A)\) has a \(Z\)-basis consisting of the \([V]\) where \(V\) is indecomposable. The Grothendieck ring \(G_0(A)\) has a basis of the \([V]\) for \(V\) simple. Various authors recently studied the Green ring of Taft algebras [\textit{H. Chen} et al., Proc. Am. Math. Soc. 142, No. 3, 765-775 (2014; Zbl 1309.16021)] and generalized Taft algebras [\textit{L. Li} and \textit{Y. Zhang}, Contemp. Math. 585, 275-288 (2013; Zbl 1309.19001)]. The paper under review studies \(r(H)\) in a still further general case, namely pointed rank one Hopf algebras \(H\) of nilpotent type for \(k\) an algebraically closed field of characteristic zero. Such an \(H\) is based on a group datum \(D=(G,X,g,u)\), where \(G\) is a finite group, \(X\) a character of \(G\) with \(u(g^n-1)=0\), \(n\) the order of \(X(g)\), and \(g\) is in the center of \(G\). \(H\) as a basis \(\{y^ih\mid h\in G,\;i=0,1,\ldots,n-1\}\). The algebra relations are \(y^n=0\) and \(yh=X(h)y\). The coalgebra structure has the \(h\) group-like and \(y\) \((1,g)\)-skew-primitive. An \(H\)-module is simple if and only if it restricts to a simple \(kG\)-module. The authors determine these, and how they decompose under tensor product. This is done by studying almost split sequences and deducing Clebsch-Gordan formulas. Finally \(r(H)\) is presented by generators and relations. \(r(H)\) is commutative and is generated by one variable over \(G_0(H)\) modulo one relation given by a Dickson polynomial multiplied by a certain almost split sequence. This recovers the results on generalized Taft algebras mentioned earlier.
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