Chevalley property and discriminant ideals of Cayley-Hamilton Hopf Algebras

Abstract

For any affine Hopf algebra $H$ which 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 $(H,C,\text{tr})$ 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 $(H,C,\text{tr})$ is non-empty then it contains the orbit of the identity element of the affine algebraic group $\text{maxSpec}C$ 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 $\text{maxSpec}C$. Then, we provide a description of the zero locus of the lowest discriminant ideal of $(H,C,\text{tr})$. It is proved that the lowest discriminant ideal of $(H,C,\text{tr})$ is of level $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))+1$, where $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$ is the Grothendieck ring of the finite-dimensional Hopf algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ and $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))$ is the Frobenius-Perron dimension of $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$. 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.