斜群范畴、与Cartan矩阵相关的代数及根格子的折叠
Skew group categories, algebras associated to Cartan matrices and folding of root lattices
AI总结：
本文研究了有限群作用下斜群范畴与Cartan矩阵代数的Morita等价关系，并通过范畴化方法实现了根格子的折叠投影。
摘要：
对于有限群作用于有限EI箭形图，我们构造其' orbifold' 商EI箭形图。与商EI箭形图相关的自由EI范畴等价于给定群作用的斜群范畴。将结果专门应用于有限群作用于有限无环箭形图的情况，我们证明，在合理条件下，路径范畴的斜群范畴等价于Cartan类型有限EI范畴。如果基域的特征为p，且作用群为p-循环群，则证明路径代数的斜群代数与定义在[C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61--158]中的Cartan矩阵代数Morita等价。我们应用Morita等价来构造与图自同构相关的根格子折叠投影的范畴化。在Dynkin情况下，范畴化对不可分解模的限制对应于正根的折叠。
英文摘要：
For a finite group action on a finite EI quiver, we construct its `orbifold' quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic $p$ and the acting group is a cyclic $p$-group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61--158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.
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