Chevalley property and discriminant ideals of Cayley–Hamilton Hopf algebras

Abstract For any affine Hopf algebra $H$ that admits a large central Hopf subalgebra, $H$ can be endowed with a Cayley–Hamilton Hopf algebra structure in the sense of De Concini–Procesi–Reshetikhin–Rosso. The category of finite-dimensional modules over any fiber algebra of $H$ is proved to be an indecomposable exact module category over the tensor category of finite-dimensional modules over the identity fiber algebra $H/\mathfrak{m}_{\overline{\varepsilon }}H$ of $H$. For any affine Cayley–Hamilton Hopf algebra $(H,C,\textrm{tr})$ such that $H/\mathfrak{m}_{\overline{\varepsilon }}H$ has the Chevalley property, it is proved that if the zero locus of a discriminant ideal of $(H,C,\textrm{tr})$ is non-empty then it contains the orbit of the identity element of the affine algebraic group $\textrm{maxSpec}C$ under the left (or right) winding automorphism group action. Its proof relies on the fact that $H/\mathfrak{m}_{\overline{\varepsilon }}H$ has the Chevalley property if and only if the $\overline{\varepsilon }$-Chevalley locus of $(H,C)$ coincides with $\textrm{maxSpec}C$. Then, we provide a description of the zero locus of the lowest discriminant ideal of $(H,C,\textrm{tr})$. It is proved that the lowest discriminant ideal of $(H,C,\textrm{tr})$ is of level $\textrm{FPdim}(\textrm{Gr}(H/\mathfrak{m}_{\overline{\varepsilon }}H))+1$, where $\textrm{Gr}(H/\mathfrak{m}_{\overline{\varepsilon }}H)$ is the Grothendieck ring of the finite-dimensional Hopf algebra $H/\mathfrak{m}_{\overline{\varepsilon }}H$ and $\textrm{FPdim}(\textrm{Gr}(H/\mathfrak{m}_{\overline{\varepsilon }}H))$ is the Frobenius–Perron dimension of $\textrm{Gr}(H/\mathfrak{m}_{\overline{\varepsilon }}H)$. Some recent results of Mi–Wu–Yakimov about lowest discriminant ideals are generalized. We also prove that all the discriminant ideals are trivial if $H$ has the Chevalley property.