Quivers, Floer cohomology, and braid group actions (Q2758962)

In this interesting paper the authors construct an injective group homomorphism from the braid group on \(n\) strings to the group of self-equivalences of the bounded derived category of a certain graded algebra \(A_n\), and as well to the group of actions on the category of Lagrangian submanifolds of some specific symplectic variety \(M_n\), the morphisms of this category being Floer cohomology groups. The latter action arises as the monodromy of the natural action of the mapping class group of the group of symplectic automorphisms of the manifold \(M_n\). The algebra \(A_n\), or related algebras appear in different contexts. The parabolic category \({\mathcal O}\) of the Lie algebra \({\mathfrak s}{\mathfrak l}_{n+1}\) is equivalent to the module category of an ungraded version of this algebra. Related to this is the discovery by Rouquier and the reviewer of self-equivalences of the bounded derived module categories of Brauer tree algebras [\textit{R. Rouquier} and \textit{A. Zimmermann}, Proc. Lond. Math. Soc., III. Ser. 87, 197-225 (2003; Zbl 1058.18007)], where one also obtains a morphism of the braid group to the group of self-equivalences of the derived category, and where one gets that the morphism is injective in case \(n= 2\) basically with image of index 4. Also there the algebra is not graded.NEWLINENEWLINE The authors show in their paper that some specific self-equivalences of the derived module category satisfy the well known braid relations. Moreover, in the image in the Grothendieck group, using the grading the authors discover the classical Bureau representation of the braid group on the Grothendieck group. The link to symplectic geometry then is given by the definition of a complex of \(A_n\)-modules associated to some class of curves corresponding to specific Lagrangian submanifolds of \(M_n\), the definition of a braid group action on these curves and the identification of the braid group action on the derived category of \(A_n\) and the braid group action on these curves. Finally, the authors define intersection numbers for that occasion and interpret the dimension of the Floer cohomology spaces in terms of these intersection numbers. The injectivity then follows from this equality.