Chevalley property of module-finite Hopf algebras and discriminant ideals

Abstract: In this paper, we study the Chevalley property of Cayley-Hamilton Hopf algebras in the sense of De Concini-Procesi-Reshetikhin-Rosso using discriminant ideals. For any affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ whose identity fiber algebra has the Chevalley property, we prove that an irreducible $H$-module $V$ has the property that $V\otimes W$ is a completely reducible $H$-module for every irreducible $H$-module $W$ if and only if $V$ is annihilated by the lowest discriminant ideal of $(H,C,\text{tr})$, which establishes a bridge between the tensor-nondegenerate behaviour of the irreducible representations of $H$ and the lowest discriminant ideal of $(H,C,\text{tr})$. Using discriminant ideals, we prove that an affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ has the Chevalley property if and only if its identity fiber algebra $H/\mathfrak{m}{\overline{\varepsilon}}H$ has the Chevalley property and all the discriminant ideals of $(H,C,\text{tr})$ are trivial, thereby resolving a question posed by Huang-Mi-Qi-Wu. Moreover, it is shown that the lowest discriminant subvariety $\mathcal{V}{\ell}$ of the algebraic group $\operatorname{maxSpec}C$ is a closed subgroup, which reflects the rigid nature of $\mathcal{V}_{\ell}$ and is effective in determining the lowest discriminant subvarieties in certain examples of low GK dimension. This rigidity property provides a method, via the lowest discriminant ideals, for constructing a large family of Hopf algebras with the Chevalley property and finite GK dimension. The results are illustrated through applications to the big quantized Borel subalgebras at roots of unity and to certain Artin-Schelter Gorenstein Hopf algebras of low GK dimension. In particular, the framework yields (non-finite) tensor categories with the Chevalley property arising from some big quantum groups at roots of unity.

• The paper precisely characterizes tensor-reducible irreducible representations with discriminant ideals, linking representation behavior and algebraic structure.
• It demonstrates that the Chevalley property holds if and only if key discriminant ideals are trivial, enabling clear rigidity and classification results.
• The work further shows that the lowest discriminant subvariety forms a smooth closed subgroup, providing new tools for quantum group analysis.

Chevalley Property, Discriminant Ideals, and the Representation Theory of Module-Finite Hopf Algebras

Overview

This paper investigates the Chevalley property for module-finite Hopf algebras in the framework of Cayley-Hamilton Hopf algebras, elucidating its deep connections with discriminant ideals. The approach leverages the powerful machinery of discriminant theory, traditionally used in the study of PI and quantum algebras, to give both structural and practical characterizations of representations whose tensor product behaves maximally nondegenerately. Several classification results and new rigidity phenomena for discriminant varieties are established, with applications to quantum groups at roots of unity and Artin-Schelter Gorenstein Hopf algebras of low GK-dimension.

Main Results

Tensor-Reducibility and Discriminant Ideals

A primary contribution of the work is the precise characterization of tensor-reducible irreducible representations in terms of the lowest discriminant and modified discriminant ideals. For an irreducible module $V$ over a Cayley-Hamilton Hopf algebra $(H,C,\mathrm{tr})$ (with $H$ module-finite over the central Hopf subalgebra $C$ and the identity fiber algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ possessing the Chevalley property), the following equivalence is established:

• $V$ is tensor-reducible (i.e., $V \otimes W$ is completely reducible for all irreducible $W$) if and only if $V$ is annihilated by the lowest discriminant ideal $D_{\ell}(H/C;\mathrm{tr})$.

Additional characterizations in terms of left/right tensor-reducibility and dual-pair complete reducibility are given. Furthermore, the connection with maximally stable modules (in the sense of Mi–Wu–Yakimov) is clarified: all maximally stable irreducibles are tensor-reducible, and conversely when the identity fiber algebra is basic, every tensor-reducible irreducible is maximally stable.

Of particular note is the explicit identification, for big quantum Borel subalgebras at roots of unity, that all tensor-reducible irreducible representations are $(H,C,\mathrm{tr})$0-dimensional.

Discriminant Ideals and Chevalley Property

The Chevalley property for module-finite Hopf algebras is shown to be entirely governed by discriminant ideals:

• $(H,C,\mathrm{tr})$1 has the Chevalley property if and only if
  • the identity fiber algebra $(H,C,\mathrm{tr})$2 has the Chevalley property,
  • and all the (modified) discriminant ideals of $(H,C,\mathrm{tr})$3 are trivial (i.e., their zero loci fill the affine algebraic group $(H,C,\mathrm{tr})$4).

This answers a question posed in previous work and sharply delineates the interplay between the algebraic structure and the category-theoretic behavior of representations.

Rigidity and Group Structure of the Lowest Discriminant Variety

A significant geometric result is the proof that the lowest discriminant subvariety $(H,C,\mathrm{tr})$5 (the vanishing set of the lowest discriminant ideal) forms a closed subgroup of the affine algebraic group $(H,C,\mathrm{tr})$6. This provides substantial additional structure: $(H,C,\mathrm{tr})$7 is smooth and equidimensional, in contrast to higher discriminant subvarieties. Moreover, via the construction of the Hopf quotient by the radical of the lowest discriminant ideal, a large new family of module-finite Hopf algebras with the Chevalley property is produced, often with infinite GK-dimension.

The method is effective in explicit settings. For example, in low GK-dimension, all possible shapes of $(H,C,\mathrm{tr})$8 are completely classified, including the realization of arbitrary finite cyclic groups as lowest discriminant subvarieties.

Technical Depth and Implications

The approach is technically intricate, combining:

• The structure theory of Cayley-Hamilton and module-finite Hopf algebras,
• Fiber module categories and their Grothendieck groups,
• Discriminant theory for noncommutative central extensions,
• Tensor categorical and Frobenius-Perron techniques for irreducible modules,
• Explicit computation for quantum groups at roots of unity, Artin-Schelter Gorenstein Hopf algebras, and generalized Taft/Liu algebras.

The main theorems highlight that discriminant ideals offer a rigorous, computable invariant capable not only of distinguishing “nondensely” reducible representations but also precisely capturing the (non-)existence of the Chevalley property in complicated noncommutative settings. The subgroup structure of lowest discriminant varieties also ties the representation theory directly to the underlying algebraic group, opening the way for further classification theorems and constructions in both finite and infinite settings.

Moreover, the explicit identification of the families of representations and subvarieties provides tools for constructing novel non-finite tensor categories with the Chevalley property arising from quantum group theory. The framework is sufficiently broad to encompass various classes of contemporary quantum and regular Hopf algebras.

Conclusion

The paper offers a comprehensive and technically robust account of discriminant ideals as controlling invariants for the Chevalley property and tensor-degeneracy phenomena in representation theory of module-finite Hopf algebras. The results clarify previous conjectures, introduce new classification and rigidity theorems, and outline a pathway for explicit construction and analysis of a wide array of examples—including those fundamental to quantum group theory and noncommutative projective geometry. The approach and outcomes provide a template for further developments in the interplay of algebraic, categorical, and geometric aspects of noncommutative algebra.