Quantum Grothendieck rings and derived Hall algebras
Abstract
Let 𝔤 = 𝔫 ⊕ 𝔥 ⊕ 𝔫- be a simple Lie algebra over ℂ of type A, D, E, and let Uq(L𝔤) 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 t-deformation 𝒦t of the Grothendieck ring of a tensor category 𝒞ℤ of finite-dimensional Uq(L𝔤)-modules. We obtain a presentation of 𝒦t by generators and relations.
Let Q be a Dynkin quiver of the same type as 𝔤. Let DH(Q) be the derived Hall algebra of the bounded derived category Db(mod(FQ)) over a finite field F, introduced by Toën [Duke Math. J. 135 (2006), 587–615]. Our presentation shows that the specialization of 𝒦t at t = |F|1/2 is isomorphic to DH(Q). Under this isomorphism, the classes of fundamental Uq(L𝔤)-modules are mapped to scalar multiples of the classes of indecomposable objects in DH(Q).
Our presentation of 𝒦t is deduced from the preliminary study of a tensor subcategory 𝒞Q of 𝒞ℤ analogous to the heart mod(FQ) of the triangulated category Db(mod(FQ)). We show that the t-deformed Grothendieck ring 𝒦t,Q of 𝒞Q is isomorphic to the positive part of the quantum enveloping algebra of 𝔤, and that the basis of classes of simple objects of 𝒦t,Q 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 Q of the Dynkin diagram, the category 𝒞Q gives a new categorification of the coordinate ring ℂ[N] of a unipotent group N with Lie algebra 𝔫, together with its dual canonical basis.
