\(\mathcal O\) algebras with three simple modules (Q1124965)

For a finite dimensional semisimple complex Lie algebra, \textit{I. N. Bernstein, I. M. Gel'fand} and \textit{S. I. Gel'fand} [Funkts. Anal. Prilozh. 10, No. 2, 1-8 (1976; Zbl 0353.18013)] defined category \(\mathcal O\) which they showed to decompose into a direct sum of module categories of finite dimensional associative algebras, the `blocks' of \(\mathcal O\). The \(\mathcal O\) algebras studied in the paper under review (motivated by work of \textit{M. J. Dyer} [see NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 95-121 (1994; Zbl 0830.20063)]) share several structural properties with the blocks of \(\mathcal O\). In particular, they are quasi-hereditary Koszul algebras. Those \(\mathcal O\) algebras are studied which have only three simple modules. Such algebras are classified by quiver and relations. Then some representation theoretic and homological properties are determined.