2025/06/27 by Huang, Yimin, Mi, Zhongkai, Qi, Tiancheng +1
#16D60 #16G30 #16T05 #17B37 #18D20 #18M05 #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2506.21879
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/\mathfrakm_εH of H. For any affine Cayley-Hamilton Hopf algebra (H,C,tr) such that H/\mathfrakm_εH has the Chevalley property, it is proved that if the zero locus of a discriminant ideal of (H,C,tr) is non-empty then it contains the orbit of the identity element of the affine algebraic group maxSpecC under the left (or right) winding automorphism group action. Its proof relies on the fact that H/\mathfrakm_εH has the Chevalley property if and only if the ε-Chevalley locus of (H,C) coincides with maxSpecC. As applications, we first provide a description of the zero locus of the lowest discriminant ideal of (H,C,tr). It is proved that the lowest discriminant ideal of (H,C,tr) is of level FPdim(Gr(H/\mathfrakm_εH))+1, where Gr(H/\mathfrakm_εH) is the Grothendieck ring of the finite-dimensional Hopf algebra H/\mathfrakm_εH and FPdim(Gr(H/\mathfrakm_εH)) is the Frobenius-Perron dimension of Gr(H/\mathfrakm_εH). Some recent results of Mi-Wu-Yakimov about lowest discriminant ideals are generalized. Secondly, we prove that all the discriminant ideals are trivial if H has the Chevalley property.