On the Cartan Graphs of Nichols Algebras over Coquasi-Hopf Algebras

Abstract

In this paper, we continue the study of the reflection theory of Nichols algebras over coquasi-Hopf algebras with bijective antipode. We prove that for a tuple of finite-dimensional simple Yetter-Drinfeld modules admitting all reflections, the associated semi-Cartan graph is actually a Cartan graph. Furthermore, we provide equivalent conditions for the finite-dimensionality of the corresponding Nichols algebra. Finally, we show that such a Cartan graph is indeed invariant under specific braided monoidal equivalences. As an application, we examine Nichols algebras of diagonal type over coquasi-Hopf algebras, proving that they yield isomorphic Cartan graphs originating from Nichols algebras of diagonal type over Hopf algebras.