Stratified algebras and classification of tilting modules

Sammanfattning: This thesis contains three papers in representation theory of algebras. It mainly studies two types of algebras; quasi-hereditary algebras and standardly stratified algebras.Paper I provides a classification of generalized tilting modules and full exceptional sequences for a family of quasi-hereditary algebras and for another related family of algebras. These algebras are referred to as leaf quotients of type A zig-zag algebras. We also give a characterization of the first family of algebras as quasi-hereditary algebras with a simple preserving duality, where exactly one indecomposable projective module is not injective.Paper II proves uniqueness of the essential order for standardly stratified algebras having a simple preserving duality. We use this result to classify, up to equivalence, regular blocks of S-subcategories in the BGG category O. We also establish some derived equivalences between blocks in type A. Additionally, the paper provides explicit formulas for the projective dimension of certain structural modules in S-subcategories of O and for the finitistic dimension of these subcategories. Paper III provides a classification of generalized tilting modules and full exceptional sequences for a family of quasi-hereditary algebras. These algebras are examples of dual extension algebras. For the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration, with respect to the relation arising from higher extensions.