AI 中文总结
研究来自\(II_1\)型因子\(Q\)自同构族的对角子因子正则性，证明其正则性与自同构类基数、子群等有关，且正则性图景是图论的，还给出了对角子因子深度准则，有助于刻画正则性。
AI 中文摘要
我们证明，当来自\(II_1\)型因子\(Q\)的有限自同构族产生的对角子因子的定义自同构类具有相同基数并形成\(\mathrm{Out}(Q)\)的子群时，该对角子因子是正则的，此子群恰好同构于该子因子的广义外尔群。此外，可以发现对角子因子正则性最清晰的图景是图论的，即当主图是完全正则平衡二分多重图时对角子因子是正则的，广义外尔群确定其大小，定义自同构的公共重数决定其正则边重数。在刻画正则性之前，我们重新审视了比施和波帕关于对角子因子标准不变量和深度的观察，给出了对角子因子具有任意规定深度的精确准则，这在正则性刻画中很有用。
英文摘要
We show that a diagonal subfactor arising from a finite family of automorphisms of a $II_1$-factor $Q$ is regular precisely when the classes of the defining automorphisms occur with same cardinality and form a subgroup of $\mathrm{Out}(Q)$, a subgroup that happens to be isomorphic to the generalized Weyl group of the subfactor. Moreover, it turns out that the cleanest picture of regularity in diagonal subfactors is graph-theoretic, namely, a diagonal subfactor is regular precisely when its principal graph is a complete, regular, balanced bipartite multigraph, with the generalized Weyl group fixing its size and the common multiplicity of the defining automorphisms determining its regular edge multiplicity. Prior to the characterization of regularity, revisiting Bisch and Popa's observations on the standard invariant and depth of diagonal subfactors, we give an exact criterion for a diagonal subfactor to have any prescribed depth, in terms of a stabilizing sequence of subsets of $\mathrm{Out}(Q)$ consisting of non-reduced alternating words in the classes of the defining automorphisms, which proves useful in the characterization of regularity.
Comments48 pages, version 2
