Fourier Transforms Of Ringel-Hall Algebras And Lusztig's Symmetries

This thesis mainly studies Fourier transforms on the Ringel-Hall algebras and their relations with Lusztig's symmetries,as well as the action of Lusztig's symmetries on finite-dimensional modules over quantum affine algebras.The thesis is divided into the following three parts:(1)Fourier transforms on the Ringel-Hall algebra of a quiver were defined by Lusztig,and he pointed out that Ringel-Hall algebras obtained by changing the orientation of a quiver are isomorphic to each other.In this part,we extend the definition of Fourier transforms to Ringel-Hall algebras of valued quivers.Since representations of valued quivers are not easy to describe,we regard them as representations of quivers with automorphism via Frobenius morphisms.By an idea of Sevenhant and Van den Bergh,we successfully give the definition of Fourier transforms on Ringel-Hall algebras of arbitrary valued quivers,and prove that they are algebra isomorphisms.As a result,it is shown that the Ringel-Hall algebra of a valued quiver is independent of its orientation.This proof is simpler than that of Deng and Xiao.(2)It has been proved independently by Sevenhant and Van den Bergh,Xiao and Yang that each BGP-reflection functor on the category of quiver representations induces an algebra isomorphism of the corresponding double Ringel-Hall algebras.By combining it with Fourier transforms,Sevenhant and Van den Bergh defined a family of automorphisms on double Ringel-Hall algebras of quivers,called SV-isomorphisms.They also proved that these isomorphisms coincide with Lusztig's symmetries on double composition subalgebras.A natural question is to determine whether the two isomorphisms agree with each other on the whole double Ringel-Hall algebras.We prove that when the quiver Q is of tree-type,the SV-isomorphism coincides with the Lusztig's symmetry up to a canonical isomorphism.We also give an example to show that this result does not hold in general.Furthermore,Sevenhant and Van den Bergh conjectured that SV-isomorphisms satisfy the braid relation as Lusztig's symmetries.This conjecture remains open since it is very difficult to describe Fourier transforms explicitly.We finally give some evidences about the conjecture in the case of the A2-type quiver.(3)Given any representation of a quantum group,we obtain a new representation which is twisted by the Lusztig's symmetry,called the twisted representation.According to Lusztig's work,each integrable representation in category O is isomorphic to its twisted representation.This part mainly studies the action of Lusztig's symmetries on finite-dimensional irreducible representations of quantum affine algebra Uv((?)n+1).Note that this class of representations does not belong to category O in general.By proving the compatibility of the evaluation homomorphism eva and Lusztig's symmetries of Uv((?)n+1),we obtain that the structure of evaluation representation twisted by a Lusztig's symmetry coincides with the original module structure.Finally,based on the results of Chari and Pressley,we prove that the twisted module structure of any finite dimensional irreducible representation is isomorphic to the original one.