Ringel-Hall algebras beyond their quantum groups I: Restriction functor and Green's formula
Abstract
In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (Lusztig,Lusztig2) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in Green and the restriction functor in Lusztig2. By a geometric way to prove the Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.
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