Degree project: The Ext-Algebra of Standard Modules of Bound Twisted Double Incidence Algebras

Abstract: Quasi-hereditary algebras are an important class of algebras with many applications in representation theory, most notably the representation theory of semi-simple complex Lie-algebras. Such algebras sometimes admit an exact Borel subalgebra, that is a subalgebra satisfying similar formal properties to the Borel subalgebras from Lie theory. This presentation is divided into two parts. In the first part, we briefly cover the definitions of quasi-hereditary algebras and exact Borel subalgebras, after which we provide a classification of quasi-hereditary algebras with two simple modules up to Morita equivalence. In the second part, given a poset X, a certain set of matrices M, and a finite set r of paths in X, we introduce an algebra A(X,M,r) called the bound twisted double incidence algebra of X. This is a generalization of the twisted double incidence algebras initially studied by Deng and Xi in [1]. We provide necessary and sufficient conditions for A(X,M,r) to be quasi-hereditary when X is a tree, and we show that A(X,M,r) admits an exact Borel subalgebra when these conditions are satisfied. We finish by providing a combinatorial description of the Ext-algebra of the standard modules of A(X,M,r).

References: [1] Bangming Deng and Changchang Xi, Quasi-hereditary algebras which are twisted double incidence algebras of posets, Beiträge zur Algebra und Geometrie, 36(1):37-71, 1995