Modified Ringel-Hall algebras and Drinfeld double
Modified Ringel-Hall algebras and Drinfeld double
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发表时间：
2016-08
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通讯作者：
Ming Lu;L. Peng
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作者：
Ming Lu;L. Peng
受Bridgeland和Gorsky关于由$\Z/2$-分次复形构造Ringel-Hall代数的工作的启发，我们考虑了遗传交换范畴$\ca$上的$\Z/2$-分次复形范畴$\cc_{\Z/2}（\ca）$，它可能没有足够的投射对象，并定义了$\ca$的修正Ringel-Hall代数，记为$\cm\ch_{\Z/2}（\ca）$，是$\cc_{\Z/2}（\ca）$的Ringel-Hall代数的商代数的局部化。我们证明了这样的Ringel-Hall代数具有一些很好的性质和结构。第一个是$\cm\ch_{\Z/2}（\ca）$有一个很好的基础。因此，$\cm\ch_{\Z/2}（\ca）$是一个在适当定义的非循环复形的量子环面上的自由模，其基由$\Z/2$-分次复形的导出范畴中的对象的同构类给出，因此在某种程度上，$\cm\ch_{\Z/2}（\ca）$具有与Gorsky的半导出Hall代数类似的意义。第二，在扭曲情形下，$\cm\ch_{\Z/2}（\ca）$同构于$\ch_{tw}^e（\ca）$的Drinfeld双Ringel-Hall代数，即$\ca$自身的扭曲扩张Ringel-Hall代数。特别地，一个有限维幂零表示的范畴和光滑射影曲线或加权射影线上的凝聚层范畴是遗传阿贝尔范畴，因此它们的扭曲修正Ringel-Hall代数与它们的Drinfeld双Ringel-Hall代数同构。最后，如果$\ca$有一个倾斜对象$T$，则它的修改的Ringel-Hall代数同构于Gorsky定义的正合范畴$\add T$的$\Z/2$-分次半导Hall代数$\cs\cd\ch_{\Z/2}（\add T）$，因此同构于$\mod（\End（T））$的Bridgeland的Ringel-Hall代数。
Inspired by the works of Bridgeland and Gorsky on constructing Ringel-Hall algebras 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 modified Ringel-Hall algebra of $\ca$, denoted by $\cm\ch_{\Z/2}(\ca)$, to be the localization of a quotient algebra of the Ringel-Hall algebra of $\cc_{\Z/2}(\ca)$. We prove such Ringel-Hall algebra to be of some nice properties and structures. The first one is that $\cm\ch_{\Z/2}(\ca)$ has a nice basis. As a consequence $\cm\ch_{\Z/2}(\ca)$ is a free module over a suitably defined quantum torus of acyclic complexes, with a basis given by the isomorphism classes of objects in the derived category of $\Z/2$-graded complexes, and so in somehow $\cm\ch_{\Z/2}(\ca)$ has a similar meaning as the semi-derived Hall algebra of Gorsky. The second one is that in twisted case $\cm\ch_{\Z/2}(\ca)$ is isomorphic to the Drinfeld double Ringel-Hall algebra of $\ch_{tw}^e(\ca)$, the twisted extended Ringel-Hall algebra of $\ca$ itself. In particular, the category of finite-dimensional nilpotent representations of a quiver and the category of coherent sheaves on a smooth projective curve or on a weighted projective line are hereditary abelian categories, and so their twisted modified Ringel-Hall algebras are isomorphic to their Drinfeld double Ringel-Hall algebras. Finally, if $\ca$ has a tilting object $T$, then its modified Ringel-Hall algebra is isomorphic to the $\Z/2$-graded semi-derived Hall algebra $\cs\cd\ch_{\Z/2}(\add T)$ of the exact category $\add T$ defined by Gorsky and so isomorphic to the Bridgeland's Ringel-Hall algebra of $\mod (\End(T))$.