On the representations of weak crossed products. (Q2909814)

All algebras in this review are finite-dimensional. For a weak Hopf algebra \(H\) for which both \(H\) and its dual \(H^*\) are semisimple, \(A\) an algebra measured by \(H\), and \(\sigma\) an invertible 2-cocycle on \(H\) with values in \(A\), the authors prove three results relating \(A\) and the weak crossed product \(A\#_\sigma H\).NEWLINENEWLINE The first is that they have the same representation dimension. The representation dimension of \(A\) is 1 if \(A\) is semisimple, otherwise it is the minimum of \(\dim.(\text{End}_A)(M)^{op}\) over all generators-cogenerators \(M\) of \(A\). -- The second is that they have the same representation type (finite, tame or wild). -- The third is that one is CM-finite \(n\)-Gorenstein if and only if the other one is. A Gorenstein algebra is a left and right Noetherian algebra with finite left and right injective dimension, and \(n\)-Gorenstein if the left injective dimension is at most \(n\). \(A\) is CM-finite (of finite Cohen-Macaulay type) if there are only finitely many isomorphism classes of indecomposable finitely-generated Gorenstein projective modules (which has to do with exact sequences of projective \(A\)-modules).