Well-filtered algebras (Q1295663)

For a quasi-hereditary algebra, the notion of standard module is well-defined as an analogue of Verma module. Finite-dimensional modules over a semisimple complex Lie algebra are known to have finite resolutions by direct sums of Verma modules [\textit{I. N. Bernstein}, \textit{I. M. Gelfand} and \textit{S. I. Gelfand}, Proc. Bolyai Math. Soc. 1971, 21-64 (1975; Zbl 0338.58019)]. Well-filtered (quasi-hereditary) algebras, which are defined in the paper under review, are characterized by the fact that every simple module has such a resolution. This happens for instance for the principal block of \({\mathfrak{sl}}(3)\), but not for larger examples [see: the reviewer, Manuscr. Math. 86, No. 1, 103-111 (1995; Zbl 0820.16011)]. Several other characterizations of well-filtered algebras are given. Moreover, for a well-filtered algebra \(A\) certain directed quotient algebras \(A^+\) and \(A^-\) are studied and their module categories are related to the category of \(A\)-modules.