Finite block theory and Hopf algebra actions. (Q2479824)

Suppose \(A\) is an algebra acted on by a Hopf algebra \(H\). The authors study the following question: when does the algebra of invariants \(A^H\) have finite block theory provided \(A\) has finite block theory? The question of when \(A\) has finite block theory provided \(A^H\) has finite block theory is also considered. A ring \(R\) has finite block theory if it can be written as a direct sum of subrings of the form \(Re_i\), where \(e_i\) is a central idempotent of \(R\) which cannot be written as a sum of central idempotents \(f\) and \(g\) such that \(fg=0\). The authors introduce condition \((N)\) on \(H\), which is satisfied if and only if whenever an \(H\) acts on a non-nilpotent algebra \(A\), \(A^H\) is non-zero. They prove that for any \(H\)-prime algebra \(R\) acted on by a finite-dimensional Hopf algebra \(H\) satisfying condition \((N)\), \(R^H\) has finite block theory and they provide a bound on the number of central idempotents of \(R^H\). Another result in the article asserts that if \(R\) is a prime algebra over a field of characteristic \(p>2\), acted on by a finite-dimensional restricted nilpotent Lie color algebra \(L=\bigoplus_{g\in G}L_g\) where \(G\) is a finite group, then \(R^L\) has finite block theory. There are some other results on the going down question. A going up result shows that for \(R\) an algebra acted on by a pointed Hopf algebra \(H\) with finitely many group-like elements, if \(R\) has infinitely many central idempotents then so does \(R^H\). Counterexamples when the hypotheses of the main results are weakened are also provided.