On the Drinfeld double of a finite group scheme and its representation category

Abstract: We classify equivalence classes of Hopf algebra quotient pairs $(D,θ)$ of the Drinfeld double $D(G)$ of a finite group scheme $G$ over an algebraically closed field $\mathbf{k}$ of characteristic $p\ge 0$, in terms of group scheme-theoretical data. We prove that such Hopf algebra quotients $D$ are Hopf algebra extensions $\mathscr{O}(K)^{\mathrm{cop}}#_σ^τ \mathbf{k}[G/H]$, where $K$ and $H$ are normal subgroup schemes of $G$ that centralize each other and $B:\mathbf{k}[H]\to \mathscr{O}(K)$ is a $G$-equivariant Hopf algebra map, and describe the surjective Hopf algebra map $θ:D(G)\twoheadrightarrow D$. Using this classification, we determine the tensor subcategories of the center $\mathscr{Z}(G):=\Rep(D(G))$ of $G$, describe their centralizers, determine when they are symmetric or non-degenerate, and give a description of their simple and projective objects using \cite{GS}. Our categorical results generalize those found in \cite{NNW} in characteristic $0$.

• The paper provides a complete Hopf-theoretic classification of quotient pairs from the Drinfeld double of finite group schemes, advancing beyond S-matrix methods.
• It establishes explicit criteria for symmetry, non-degeneracy, and Lagrangianity in the representation category, detailing the roles of centralizing subgroup schemes and equivariant maps.
• The framework generalizes classical results to include non-semisimple and positive characteristic cases, paving the way for applications in quantum groups and modular representation theory.

Drinfeld Double of Finite Group Schemes and Their Representation Categories

Overview

This paper provides a comprehensive classification of Hopf algebra quotient pairs $(D,\theta)$ of the Drinfeld double $D(G)$ of a finite group scheme $G$ over an algebraically closed field of arbitrary characteristic. It replaces the classical $S$-matrix-centric approaches with an entirely Hopf-theoretic framework that applies equally in both zero and positive characteristic settings, including cases where $D(G)$ fails to be semisimple. The structural implications of this classification are used to fully describe the braided tensor subcategory lattice of the representation category $\mathscr{Z}(G) = Rep(D(G))$, establish explicit criteria for symmetry and non-degeneracy, and provide detailed group scheme-theoretical data for simple and projective objects in these categories. Additionally, the construction generalizes previous results established in characteristic zero and paves the way for applications to twisted Drinfeld doubles.

Hopf Algebra Quotients of Drinfeld Doubles

The central technical achievement is the construction and classification of Hopf algebra quotient pairs of $D(G)$. The main results demonstrate that given any pair of normal subgroup schemes $K,H \subset G$ that centralize each other, along with a $G$-equivariant Hopf algebra map $B: \mathbf{k}[H] \to \mathscr{O}(K)$, one obtains a Hopf algebra quotient of $D(G)$0:

$D(G)$1

with explicit composition rules determined by the cocycle and co-cocycle data $D(G)$2 arising naturally from the group scheme and Hopf algebra structure. The surjective Hopf algebra map $D(G)$3 is characterized precisely, with kernel described by explicit generators involving the augmentation ideal and the $D(G)$4-equivariant map $D(G)$5.

The construction recovers as special cases prior results in characteristic zero and in the constant group case, and produces large new families of ribbon braided and quasitriangular (and, under additional conditions, triangular or factorizable) Hopf algebras as quotients of $D(G)$6. The classification is shown to be complete and unambiguous, with equivalence classes of quotient pairs corresponding bijectively to the triple $D(G)$7.

Tensor Subcategories of $D(G)$8

With the quotient theory established, the paper proceeds to classify tensor subcategories of $D(G)$9. It is shown that each tensor subcategory corresponds uniquely to a triple $G$0 as above, and the lattice of such subcategories is explicitly described in terms of inclusion relations between normal subgroup schemes and compatibility of the equivariant morphisms.

The categorical structure, including direct sum decompositions, centralizers, and inclusion relations, is made explicit and described through group scheme-theoretical data. The assignments $G$1 yield ribbon braided tensor subcategories whose Frobenius-Perron dimension is $G$2.

M\"uger Centralizer and Non-Degeneracy Criteria

The paper identifies the M\"uger centralizer of any tensor subcategory $G$3 as $G$4, with $G$5 a dual $G$6-equivariant map. This result is verified by explicit computation of the image of $G$7 under the relevant quotient maps, establishing the centralizer in the sense of braided tensor category theory.

Sharp criteria for symmetry (triangularity), non-degeneracy (factorizability), and Lagrangianity are formulated entirely in terms of group-theoretical and pairings:

• Symmetry: $G$8 and $G$9.
• Non-degeneracy: $S$0 and a certain pairing is a Hopf algebra isomorphism, implying $S$1 is self-dual.
• Lagrangianity: $S$2 and $S$3, which forces commutativity and identifies the quotient as a group algebra.

Simple and Projective Objects

Simple and projective objects within $S$4 are described in detail via group scheme-theoretical constructions and explicit corepresentation theory. The paper employs results from [GS] to provide explicit formulas for simple and projective modules, their dimensions, and decomposition properties. The classification yields a stratification of the category corresponding to conjugacy classes in $S$5 and connects modular and classical representation theory.

Examples and Special Cases

The theory is instantiated in several key cases:

• Constant groups: Classical group algebra and module paradigms are recovered, with the subcategory structure matching that in [NNW].
• Connected group schemes: The role of restricted Lie algebras and local commutative Hopf algebra quotient pairs is elucidated.
• Commutative and self-dual group schemes: Symmetry and Lagrangianity criteria become transparent, with explicit description of the relevant group scheme extensions.
• Cases with trivial or non-trivial cocycle data: Both tensor product and more exotic structures are constructed, including explicit non-trivial co-cocycle examples.

Implications and Outlook

The results establish precise group scheme-theoretical control over the representation theory of Drinfeld doubles in both semisimple and non-semisimple contexts. The classification of tensor subcategories, their centralizers, and the structure of simple and projective objects lays groundwork for further investigation in finite braided tensor categories, Hopf-Galois theory, and modular representation theory.

Practically, these results may inform the construction and classification of quantum groups, invariants in braided tensor categories, and applications in modular representation theory and algebraic geometry, particularly in positive characteristic. The theoretical structure is sufficiently general to be applicable to quasi-Hopf algebras and twisted Drinfeld doubles, suggesting significant further developments are likely in the study of finite algebraic group schemes and their representation categories.

Conclusion

This paper provides an exhaustive Hopf-theoretic framework for understanding the representation categories of the Drinfeld double of a finite group scheme. The fine classification of quotient pairs, tensor subcategories, and their categorical properties advances the structure theory for finite braided tensor categories beyond the limitations of semisimplicity and characteristic zero. The explicit, group-theoretical nature of the constructions makes them amenable to further generalization, including applications to twisted and quasi-Hopf structures, and sets a clear direction for the exploration of more general quantum symmetries in algebra and geometry.

(2603.29639)