Quantised algebras and quiver representation theory

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Quiver representation theory is a rich research subject and has broad connections to various subjects in mathematics. This project focuses on its connections to quantised algebras and Lie algebras. In early 1990's Ringel defined a Hall algebra on the categories of representations of quivers and showed such an algebra gives a realisation of the positive part of quantised enveloping algebras, i.e. quantum groups, extending Gabriel and Kacs results on the connections of quivers and Lie algebras. Ringel's remarkable work gives a meaning of the multiplication in quantised enveloping algebras and introduces a powerful tool into the quantum world. It has inspired a series of distinguished works in quantised algebras, using the Hall algebra approach. Also in 1990s, Lusztig gave a geometric construction of q-Schur algebras with his collaborators Beilinson and MacPherson. He also constructed the whole quantised enveloping algebras, using perverse sheaves. In 2011, Bridgeland gave a new construction of the whole quantised enveloping algebra, on a more elementary level than Lusztig's construction. This project will in particular explore new constructions of affine quantised enveloping algebras, their relations to quantised algebras, their degenerate version at zero and new connections between quiver representation theory and Lie theory.