Structure and representations of certain classes of infinite-dimensional algebras

We study several infinite-dimensional algebras and their representation theory. In Paper I, we study the category O for the (centrally extended) Schrödinger Lie algebra, which is an analogue of the classical BGG category O. We decompose the category into a direct sum of "blocks", and describe Gabriel quivers of these blocks. For the case of non-zero central charge, we in addition find the relations of these quivers. Also for the finite-dimensional part of O do we find the Gabriel quiver with relations. These results are then used to determine the center of the universal enveloping algebra, the annihilators of Verma modules, and primitive ideals of the universal enveloping algebra which intersect the center of the Schrödinger algebra trivially. In Paper II, we construct a family of path categories which may be viewed as locally quadratic dual to preprojective algebras. We prove that these path categories are Koszul. This is done by constructing resolutions of simple modules, that are projective and linear up to arbitrary position. This is done by using the mapping cone to piece together certain short exact sequences which are chosen so as to fall into three managable families. In Paper III, we consider the category of injections between finite sets, and also the path category of the Young lattice subject to the relations that two boxes added to the same column in a Young diagram yields zero. We construct a new and direct proof of the Morita equivalence of the linearizations of these categories. We also construct linear resolutions of simple modules of the latter category, and show that it is quadratic dual to its opposite. In Paper IV, we define a family of algebras using the induction and restriction functors on modules over the dihedral groups. For a wide subfamily, we decompose the algebras into indecomposable subalgebras, find a basis and relations for each algebra, as well as explicitly describe each center.