Quantum Grothendieck rings and derived Hall algebras

Let g = n ⊕ h ⊕ n be a simple Lie algebra over C of type , , , and let g) be the associated quantum loop algebra. Following Nakajima [Ann. of Math. (2) 160 (2004), 1057–1097], Varagnolo–Vasserot [Studies in memory of Issai Schur, Birkhäuser-Verlag, Basel (2002), 345–365], and the first author [Adv. Math. 187 (2004), 1–52], we study a -deformation K of the Grothendieck ring of a tensor category C of finite-dimensional g)-modules. We obtain a presentation of K by generators and relations. Let be a Dynkin quiver of the same type as g. Let DH( ) be the derived Hall algebra of the bounded derived category (mod( )) over a finite field , introduced by Toën [Duke Math. J. 135 (2006), 587–615]. Our presentation shows that the specialization of K at = | is isomorphic to DH( ). Under this isomorphism, the classes of fundamental g)-modules are mapped to scalar multiples of the classes of indecomposable objects in DH( ). Our presentation of K is deduced from the preliminary study of a tensor subcategory C of C analogous to the heart mod( ) of the triangulated category (mod( )). We show that the -deformed Grothendieck ring K of C is isomorphic to the positive part of the quantum enveloping algebra of g, and that the basis of classes of simple objects of K corresponds to the dual of Lusztig's canonical basis. The proof relies on the algebraic characterizations of these bases, but we also give a geometric approach in the last section. It follows that for every orientation of the Dynkin diagram, the category C gives a new categorification of the coordinate ring C[ ] of a unipotent group with Lie algebra n, together with its dual canonical basis.