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.