Semi-derived Ringel-Hall algebras and Drinfeld double

摘要

Inspired by the works of Bridgeland and Gorsky on constructing Ringel-Hallalgebras from $Z/2$-graded complexes, we consider the category$cc_{Z/2}(ca)$ of $Z/2$-graded complexes over a hereditary abelian category$ca$ which may not have enough projective objects, and define the modifiedRingel-Hall algebra of $ca$, denoted by $cmch_{Z/2}(ca)$, to be thelocalization of a quotient algebra of the Ringel-Hall algebra of$cc_{Z/2}(ca)$. We prove such Ringel-Hall algebra to be of some niceproperties and structures. The first one is that $cmch_{Z/2}(ca)$ has anice basis. As a consequence $cmch_{Z/2}(ca)$ is a free module over asuitably defined quantum torus of acyclic complexes, with a basis given by theisomorphism classes of objects in the derived category of $Z/2$-gradedcomplexes, and so in somehow $cmch_{Z/2}(ca)$ has a similar meaning as thesemi-derived Hall algebra of Gorsky. The second one is that in twisted case$cmch_{Z/2}(ca)$ is isomorphic to the Drinfeld double Ringel-Hall algebraof $ch_{tw}^e(ca)$, the twisted extended Ringel-Hall algebra of $ca$ itself.In particular, the category of finite-dimensional nilpotent representations ofa quiver and the category of coherent sheaves on a smooth projective curve oron a weighted projective line are hereditary abelian categories, and so theirtwisted modified Ringel-Hall algebras are isomorphic to their Drinfeld doubleRingel-Hall algebras. Finally, if $ca$ has a tilting object $T$, then itsmodified Ringel-Hall algebra is isomorphic to the $Z/2$-graded semi-derivedHall algebra $cscdch_{Z/2}(dd T)$ of the exact category $dd T$ definedby Gorsky and so isomorphic to the Bridgeland's Ringel-Hall algebra of $mod(End(T))$.