Frobenius-Perron dimensions of integral \(\mathbb{Z}_+\)-rings and applications (Q2198620)

The goal of this paper is to define Frobenius-Perron dimension \({(\operatorname{FPdim})}\) for an integral \(\mathbb{Z}_+\)-ring \(A\) that is not necessarily fusion and to give applications of this notion.\newline Several properties are proven for \(\operatorname{FPdim}(A)\). Among these is a theorem which states that if \(A\) has rank \(r\) and \(X\in A\) is a \(\mathbb{Z}_+\)-generator, then \((2\operatorname{FPdim}(X))^{r-1}\geq \operatorname{FPdim}(A)\). \newline This Frobenius-Perron dimension leads to the following results, proven for any non-semisimple quasi-Hopf algebra \(H\) over \(\mathbb{K}\) with two simple modules. Let \(\mathcal{C}=\operatorname{Rep}(H)\), \(C\) be the Cartan matrix of \(\mathcal{C}\), and \(X\) be a \(\mathbb{Z}_+\)-generator of \(\mathcal{C}\). Then \(|\det{C}|\leq\left(\frac{\operatorname{FPdim}(\mathcal{C})}{2\operatorname{FPdim}(X)}\right)^2\). Moreover, if \(\mathcal{C}\) is not pointed, \(Id \cong**\) as an additive functor, and the characteristic of \(\mathbb{K}\) is either \(0\) or \(p > \left(\frac{\operatorname{FPdim}(\mathcal{C})}{2\operatorname{FPdim}(X)}\right)^2\), then \(\operatorname{FPdim}(\mathcal{C})\) is not square-free. Additional results are given depending on the decomposition of \(X^2\).\newline Another application in this work is in the study of Hopf algebras \(H\) of prime dimension \(p\) over a field of characteristic \(q\). Let \(\phi(x)= 2^{-2/3} \exp\left(\frac{1}{2} W (2^{13/3} \log x\right)\), with \(W\) the Lambert \(W\)-function. The author proves that if \(H\) is not both commutative and cocommutative, then \begin{align*} \frac{p}{q+2}>\max\left( \frac{14}{3},\min\left(9,\phi(p)\right)\right). \end{align*}