Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 36 (1995), No. 1

Quasi--hereditary Algebras which are Twisted Double Incidence Algebras
of Posets

Bangming Deng and Changchang Xi

For each finite poset, we associate it with a family M of matrices
and define the corresponding M--twisted double incidence  algebra which is a
generalization of the construction given by Dyer.  In the  paper we mainly
study the quadratic dual and Ringel  dual of the M--twisted double incidence
algebra.  We prove that if the given poset is a tree then its M-twisted double
incidence algebra is a BGG--algebra and the Ringel dual can  be determined in
detail (with some natural restriction on  matrices). Moreover, we show that
in  the tree case with all matrices non--zero the processes of forming Ringel
duals and quadratic duals of M--twisted double incidence algebras,
respectively, commute with each other.