关于余拟霍普夫代数上Nichols代数的Cartan图
On the Cartan Graphs of Nichols Algebras over Coquasi-Hopf Algebras
AI总结：
研究余拟霍普夫代数上Nichols代数反射理论，证明特定条件下半Cartan图是Cartan图，给出Nichols代数有限维等价条件，表明Cartan图在特定辫幺半等价下不变，并应用于对角型Nichols代数。
AI中文摘要：
本文继续研究具有双射对极的余拟霍普夫代数上Nichols代数的反射理论。证明对于一组允许所有反射的有限维单Yetter-Drinfeld模，相关的半Cartan图实际上是一个Cartan图。此外，我们给出了相应Nichols代数有限维性的等价条件。最后，我们表明这样的Cartan图在特定的辫幺半等价下确实是不变的。作为应用，我们研究了余拟霍普夫代数上对角型的Nichols代数，证明它们产生与霍普夫代数上对角型Nichols代数同构的Cartan图。
英文摘要：
Over an algebraically closed field of characteristic zero, let $H$ be a coquasi-Hopf algebra with bijective antipode, and let $M$ be a tuple of finite-dimensional simple Yetter--Drinfeld modules over $H$. We prove that, if $M$ admits all reflections, then its associated semi-Cartan graph is a Cartan graph. We characterize the finiteness of this Cartan graph by tensor decomposability of $\mathcal B(M)$ and obtain a finite-dimensionality criterion of Nichols algebras. We also show that braided monoidal equivalences preserve reflections and the associated Cartan graphs. As applications, we prove that every Cartan graph associated with a diagonal type tuple over a finite abelian group equipped with an abelian \(3\)-cocycle is covered by one arising from a diagonal type tuple over some finite abelian group $G$ with trivial associator, and that their real-root sets agree at corresponding objects. We also construct a Cartan graph of the former kind that cannot be obtained from any diagonal tuple in \({}_G^G\mathcal{YD}\).
