Homological structures at the interface of abstract representation theory and algebraic Lie theory

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This project will investigate homological structures of algebras at the interface of abstract representation theory of finite dimensional algebras and algebraic Lie theory. The main topics will be (a) homological dimensions such as representation dimension, dominant dimension and finitistic dimension, (b) homological conjectures such as finitistic dimension conjecture, Nakayama’s conjecture, Cartan determinant conjecture and strong no loops conjecture, (c) structures of algebras such as finite dimensional and affine cellular and quasi-hereditary structures and Borel subalgebras and bocses, (d) equivalences of abelian and triangulated categories, tilting objects and recollements. Particular attention will be paid to connections and interactions between these topics. The results will be tested on and applied to algebras of interest in algebraic Lie theory such as Schur algebras of classical groups, blocks of the Bernstein-Gelfand-Gelfand category O, Brauer algebras, group algebras of symmetric groups and affine Hecke algebras.