Frobenius morphisms and representations of algebras
Frobenius morphisms and representations of algebras
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DOI：
10.1090/s0002-9947-06-03812-8
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发表时间：
2003-07
影响因子：
1.3
通讯作者：
B. Deng;J. Du
中科院分区：
文献类型：
--
作者：
B. Deng;J. Du
通过在q个元素的有限域F q 的代数闭包F q 上引入代数A上的Frobenius态射F及其模，我们建立了A在F q 上的表示论与F-不动点代数A F 在F q 上的表示论之间的关系。更准确地说，我们证明有限维 A F 模的范畴 mod-A F 等价于有限维 F 稳定 A 模的子范畴，并且当 A 是有限维时，我们在不可分解 A F 模的同类和不可分解 A 模同类的 F 轨道之间建立双射。将该理论应用于具有自同构的箭袋表示，我们表明 F q 上的调制箭袋（或物种）的表示可以解释为 F q 上相应箭袋的 F 稳定表示。我们进一步证明 F q 上的每个有限维遗传代数都是 Morita 等价于某个 A F ，其中 A 是 F q 上的颤动 Q 的路径代数，F 是由 Q 的某种自同构导出的。 A 和 A F 的 Auslander-Reiten 理论之间建立了密切的关系。特别是，我们证明了 A F 的 Auslander-Reiten（调制）箭袋是通过“折叠”A 的 Auslander-Reiten 箭袋获得的。最后，通过采用 Frobenius 不动点，我们能够计算具有给定维度向量的 F q 上调制箭袋的不可分解表示的数量，并将 Kac 定理推广到所有调制箭袋及其相关的 Kac-Moody由可对称广义嘉当矩阵定义的代数。
By introducing Frobenius morphisms F on algebras A and their modules over the algebraic closure F q of the finite field F q of q elements, we establish a relation between the representation theory of A over F q and that of the F-fixed point algebra A F over F q . More precisely, we prove that the category mod-A F of finite-dimensional A F -modules is equivalent to the subcategory of finite-dimensional F-stable A-modules, and, when A is finite dimensional, we establish a bijection between the isoclasses of indecomposable A F -modules and the F-orbits of the isoclasses of indecomposable A-modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over F q can be interpreted as F-stable representations of the corresponding quiver over F q . We further prove that every finite-dimensional hereditary algebra over F q is Morita equivalent to some A F , where A is the path algebra of a quiver Q over F q and F is induced from a certain automorphism of Q. A close relation between the Auslander-Reiten theories for A and A F is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of A F is obtained by "folding" the Auslander-Reiten quiver of A. Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over F q with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.