Maximal quotient algebra of a Hopf-module algebra
Let \(H\) be a finite dimensional semisimple Hopf algebra and \(A\) be a left \(H\)-module algebra. Denote by \(I\) the injective hull of \(A\) as the left \(A\#H\)-module. Put \(T=\text{End}_ AI\), \(Q=\text{End}_ TI\). A left ideal \(J\) in \(A\) is \(H\)-dense if \(Jt=0\) implies \(At=0\) for any \(t\in T\). A left ideal \(J\) is dense if the right annihilator of \(Ja^{-1}\) in \(A\) is equal to zero for any \(a\in A\). Theorem 1. A left ideal \(J\) in \(A\) is dense if and only if it is \(H\)- dense. -- Let \({\mathcal F}_ H\) be a filter of \(H\)-dense left ideals in \(A\). Theorem 2. \(Q\simeq A_{{\mathcal F}_ H}\), where \(A_{{\mathcal F}_ H}\) is Martindale's quotient ring with respect to \({\mathcal F}_ H\). -- Theorem 3. Consider an embedding of \(H\)-module algebras \(A\subseteq S\) such that \(A\) is an essential \(A\# H\)-submodule of \(S\) and \(As^{-1}\) is an \(H\)-dense ideal for any \(s\in S\setminus 0\). Then there exists a unique embedding \(S\to Q\) of \(H\)-module algebras. -- It follows that the action of \(H\) on \(A\) is uniquely extended to the left maximal quotient algebra \(Q\). If \(A\) is left non-singular and either \(\text{tr}:Q\to Q^ H\) is non-degenerate or \(A\) has ACC on direct sums of left ideals, then \(Q(A^ H)=Q(A)^ H\). Similar problems are discussed in Ph.D. dissertation by \textit{A. Sidorov} [Novosibirsk, 1989], see also [\textit{S. Montgomery}, Hopf algebras and their actions on rings, CBMS, No. 82 (1993), \S 6.4].
finite dimensional semisimple Hopf algebras
scientific article; zbMATH DE number 738963
scientific article