Lattice theory of torsion classes
Lattice theory of torsion classes
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发表时间：
2017-11
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通讯作者：
Laurent Demonet;O. Iyama;Nathan Reading;I. Reiten;H. Thomas
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作者：
Laurent Demonet;O. Iyama;Nathan Reading;I. Reiten;H. Thomas
对于域$k$上的有限维代数$A$，我们考虑扭类的完备格$\算子名{\mathsf{tors}}A$.我们引入了它的Hasse箭图的砖块标号，并利用它研究了$操作符名称{\mathsf{tors}}A$的格同余.特别地，我们给出了所谓强迫序的表象理论解释。当$I$是$A$的双边理想时，$\算子名{\mathsf{tors}}(A/I)$是$\算子名{\mathsf{tors}}A$的格商，称为代数商，相应的格同余称为代数同余.本文的第二部分研究了代数同余。我们刻画了由代数同余收缩的$\算符名称{\mathsf{tors}}A$的Hasse箭图的箭头。在第三部分中，我们详细研究了预射影代数$\PI$的情形，其中$\算子名{\mathsf{tors}}\Pi$是赋有弱序的Weyl群。特别地，当$q$是动态箭图时，我们给出了$\操作符名称{\mathsf{tors}}kq$与寒武纪格同构的一个新的证明，这更具表示理论意义。我们还证明了在$A$型中，$操作符名称{\mathsf{tors}}\pI$的代数商正是它的Hasse-正则格商。
For a finite-dimensional algebra $A$ over a field $k$, we consider the complete lattice $\operatorname{\mathsf{tors}} A$ of torsion classes. We introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of $\operatorname{\mathsf{tors}} A$. In particular, we give a representation-theoretical interpretation of the so-called forcing order. When $I$ is a two-sided ideal of $A$, $\operatorname{\mathsf{tors}} (A/I)$ is a lattice quotient of $\operatorname{\mathsf{tors}} A$ which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of $\operatorname{\mathsf{tors}} A$ that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras $\Pi$, for which $\operatorname{\mathsf{tors}} \Pi$ is the Weyl group endowed with the weak order. In particular, we give a new proof of the isomorphism between $\operatorname{\mathsf{tors}} k Q$ and the Cambrian lattice when $Q$ is a Dynkin quiver, which is more representation theoretical. We also prove that, in type $A$, the algebraic quotients of $\operatorname{\mathsf{tors}} \Pi$ are exactly its Hasse-regular lattice quotients.