Hall Algebras Associated To Derived Categories And Braid Group Actions

We consider the Hall algebras associated to the derived categories,especiallythe derived Hall algebra defined by Toe¨n[47]. It is well known that the Hall algebraprovides a beautiful model for the realization of quantum groups. We are interested inthe relationship between derived Hall algebras and quantum groups. The derived Hallalgebra can be seen as a Z-periodic model for quantum groups. Moreover, some resultson quantum groups can be extended to the derived Hall algebras. The main results weobtained are as follows.Let k be a finite field. Firstly, we consider the derived category Db(A ) of a hered-itary k-category A . There are two important results on extending the Hall algebraformalism to Db(A ). One is the lattice algebra L(A ) defined by Kapranov[23]. Theother is the derived Hall algebra DH(A ) defined by Toe¨n. We prove that the derivedHall algebra can be identified with the lattice algebra by the"twist and extend"proce-dure, with a suitable subalgebra closely related to the Heisenberg double.Secondly, letΛbe a finite dimensional hereditary k-algebra. There is a canonicalisomorphism between the positive part U+ of the quantum group Uq(g) and the Hallalgebra H(Λ), if the simple Lie algebra g andΛenjoy a common Dynkin diagram. Itis a natural question to ask how to decompose the root vectors into linear combina-tions of monomials of Chevalley generators of U+. We ask a similar question for thederived Hall algebra DH(Λ): how to decompose the exceptional elements into linearcombinations of monomials of simple generators of DH(Λ)? We answer this questionby applying two algorithms, respectively induced by the braid group action on the ex-ceptional sequences in the derived category and the structure of the Auslander-Reitenquiver ofΛ. This is a natural generalization of Chen and Xiao's algorithms[6, 7]. Wealso observe that there are many quantum Serre relations hidden inside DH(Λ).Thirdly, consider a valued quiver Q without oriented cycles and a reduced k-species S of type Q. LetΛbe the tensor algebra of S . We can identify the cate-gory rep S with modΛ, and denote the derived category of rep S by Db(S ). Forany sink or source i of Q, we give a direct definition of the BGP re?ection functor σi±: Db(S )→Db(σiS ) (compare [51]). It is an equivalence of derived categories,so induces an isomorphism between extended derived Hall algebras. We can also setup canonical isomorphisms between extended derived Hall algebras using the structureof Heisenberg double. Thus we can extend Lusztig's symmetries[25] to the extendedderived Hall algebra: Ti,Ti : DH (Λ)→DH (Λ). The braid group relations areverified in the cases of rank 2 and ADE type.