"This project will investigate the structure and representation theory of quantized enveloping algebras of finite dimensional semisimple complex Lie algebras and of affine Kac-Moody algebras, and of other algebras closely related to them: Hall algebras and (quantized) Schur algebras. The project's starting point is a construction, due to Beilinson, Lusztig and MacPherson, of an epimorphism from a type A quantum group onto a Schur algebra. So far, this construction has been limited to type A (with linear orientation on the positive part). The project aims at extending this result to general Dynkin or even affine (extended Dynkin) type. The project will take the following novel approach. At the beginning, it will restrict to the positive parts of quantized en veloping algebras. These can be identified with Hall (affine type: composition) algebras of quivers. Then representation theory (Auslander Reiten theory and covering theory) and geometry of quivers will be used to define and construct positive Schur algebr as quotients and to describe the kernels of these epimorphisms. Using representation theory of finite dimensional algebras and homological algebra, the structure of positive Schur will then be described. Using Drinfeld double constructions and derived c ategories (especially root categories), the information obtained will then be transferred from positive to full Schur algebras and to quantum groups. Quantum groups and Schur algebras have a wide range of applications, ranging from Lie algebras and algebr aic groups or symmetric groups and finite groups of Lie type over knot theory, homotopy theory, K-theory and functor cohomology to mathematical physics and quantum field theory. The project will extend a fundamental technique in this area and it is designed to be part of a long-term and large-scale project, with a range of abstract and computational applications."
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