Kac–Paljutkin Hopf Algebra

• The Kac–Paljutkin Hopf algebra is an 8-dimensional semisimple Hopf algebra that is both noncommutative and noncocommutative, serving as a cornerstone in quantum symmetry research.
• Its representation category, equivalent to a Tambara–Yamagami category, exhibits fusion rules that highlight unique non-group theoretical behaviors.
• It demonstrates categorical rigidity against cocycle deformations and fosters generalizations that deepen insights into quantum topology and invariant theory.

The Kac–Paljutkin Hopf algebra occupies a foundational position as the first finite-dimensional example exhibiting both noncommutativity and noncocommutativity. Its algebraic, categorical, and representation-theoretic properties have profoundly shaped the paper of non-group-type semisimple Hopf algebras, quantum symmetries, and the interplay between algebraic and categorical invariants. The Kac–Paljutkin algebra is now understood within a broader landscape of generalizations, deformation phenomena, Galois-theoretical rigidity, and connections to fusion categories, quantum topology, and tensor categorical invariants.

1. Algebraic Structure and Presentation

The original Kac–Paljutkin algebra, denoted $H_8$, is the unique (up to isomorphism) 8-dimensional semisimple Hopf algebra that is neither commutative nor cocommutative, classically presented by Kac and Paljutkin in the 1960s. It is defined over a field $\mathbb{k}$ (char $\mathbb{k}=0$).

Generators and Relations. The algebra is generated by $x$, $y$, $z$ subject to

$$x^2 = y^2 = 1, \quad xy = yx, \quad z^2 = \frac{1}{2}(1 + x + y - xy), \quad zx = yz, \quad zy = xz.$$

The coalgebra structure is

$$\Delta(x) = x \otimes x,\quad \Delta(y) = y \otimes y, \quad \Delta(z) = J(z \otimes z), \quad \varepsilon(x) = \varepsilon(y) = \varepsilon(z) = 1, \quad S(z) = z.$$

Here, $J$ is a twist element (see (Pansera, 2017)), and the noncocommutativity is encoded in the comultiplication for $z$.

Bicrossed Product and Group-Theoretical Construction. $H_8$ admits a bicrossed product description $H_8 \cong kF\#_{\sigma}kT$ with $F \cong \mathbb{Z}_2 \times \mathbb{Z}_2$, $T \cong \mathbb{Z}_2$, and a nontrivial matched pair (nontrivial action of $T$ on $F$), together with a 3-cocycle in the abelian extension (see (Castaño et al., 2016)).

Basis and Representation Theory. The algebra’s basis consists of the four group-like elements $\{1,x,y,xy\}$ and four additional elements (built from $z$ and its products with $x, y, xy$), reflecting its structure as a semisimple, non-pointed Hopf algebra.

2. Representation Category and Fusion Rules

The finite-dimensional representation category of $H_8$ encapsulates fusion behavior beyond group and pointed Hopf algebras. It consists of four one-dimensional irreducible representations and one irreducible representation of dimension two, with fusion rules corresponding to the Tambara–Yamagami category of the Klein 4-group $K_4 = \mathbb{Z}_2 \times \mathbb{Z}_2$ (see (Kitagawa, 2019, Goswami et al., 9 Oct 2025)).

Tambara–Yamagami Structure. The corepresentation category $\operatorname{Rep}(H_8)$ is tensor equivalent to the Tambara–Yamagami category $\mathcal{TY}(K_4,\chi,\tau)$:

• Simple objects: $K_4 \cup \{\rho\}$,
• Fusion: $s \otimes \rho \cong \rho \cong \rho \otimes s$ for $s\in K_4$; $\rho \otimes \rho \cong \bigoplus_{s\in K_4} s$,
• Associativity data determined by the symmetric bicharacter $\chi$ and a parameter $\tau$ (e.g., $\tau = \frac{1}{2}$ for $H_8$).

Significance. This fusion category, characterized by non-invertible objects and non-group-theoretical fusion, is critical in demonstrating that $H_8$ is not equivalent, as a tensor category, to the representation category of any group algebra or its dual.

3. Categorical Rigidity, (Co)Cocycle Deformations, and Morita Invariants

A distinctive feature of $H_8$ is the categorical rigidity with respect to cocycle deformations, Galois objects, and Morita equivalence (Castaño et al., 2016, Wakui, 2018, Goswami et al., 9 Oct 2025).

3.1 Galois Objects and Cocycle Deformations

• Galois Objects: All right $H_8$-Galois objects are trivial; the only fiber functor on the corresponding group-theoretical fusion category arises from the trivial subgroup ((Castaño et al., 2016), Thm. 2.2).
• Cocycle Deformations: There are no nontrivial 2-cocycle deformations, i.e., any cocycle twist yields a Hopf algebra isomorphic to $H_8$. Formally, for all cocycles $\omega$, $H_8^{\omega}\cong H_8$.
• Implication: The representation category is categorically rigid—cocycle deformation cannot connect $H_8$ to any other (non-isomorphic) semisimple Hopf algebra, nor can it yield new monoidal equivalences.

3.2 Categorical Morita Equivalence

• Classification: $H_8$ defines its own categorical Morita equivalence class; its Drinfeld center is not braided equivalent to that of any group algebra or dual, nor to those of the semisimple Hopf algebras $A_{(c,g)}$ of other dimensions ((Castaño et al., 2016), Appendix B).
• Significance: This demarcates $H_8$ and its analogues as isolated points in the landscape of semisimple Hopf algebras under all known Morita-type equivalence relations.

3.3 Braided Morita Equivalence

• Classification: Braided Hopf algebra structures on $H_8$ fall into exactly six braided Morita equivalence classes, distinguished by polynomial invariants and coribbon elements ((Wakui, 2018), Theorem 5.3).
• Invariants: The key invariants are Drinfeld elements, polynomial invariants $P_{A,R}^{(d)}(x)$ (Equation 2.2), and coribbon structure $\operatorname{CRib}(A,\sigma)$.
• Automorphisms: $$\operatorname{Aut}(H_8)\cong\mathbb{Z}_2 \oplus \mathbb{Z}_2$$ acts on this space of braidings, permuting equivalence classes.

4. Generalizations: Higher-Dimensional Kac–Paljutkin-Type Hopf Algebras

Significant work has been devoted to generalizing $H_8$ to higher dimensions (see (Chen et al., 2018, Lomp, 1 May 2025, Bottegoni et al., 3 Dec 2024)).

• Family $H_{2n^2}$: Introduced as semisimple Hopf algebras of dimension $2n^2$ (Pansera); $H_8$ is the case $n=2$. The extension to $H_{n,m}$ of dimension $n^m m!$ has been recently constructed (Lomp, 1 May 2025).
  • Algebraic realization: $$H_{n,m}=\mathbb{K}[\mathbb{Z}_n]^{\otimes m}\#_{\gamma}\Sigma_m$$, as crossed products with a symmetric group, via explicit cocycle twisting by elements $J$ built from central idempotents.
  • Irreducible Representations: Families of $m$-dimensional irreducible, inner-faithful representations are constructed.
• Grothendieck Ring: For $H_{2n^2}$, all irreducibles are classified and the representation ring is described by generators and relations, generalizing the Kac–Paljutkin case (Chen et al., 2018).
• Infinitesimal R-Matrices and Rigidity: For $n\ge3$, $H_{2n^2}$ admits only the trivial infinitesimal $\mathcal{R}$-matrix, precluding interesting formal quasitriangular deformations (Bottegoni et al., 3 Dec 2024).

5. Module Categories, Partial Comodules, and Nichols Algebras

$H_8$ links representation theory and the classification of module and comodule categories.

5.1 Simple Yetter–Drinfeld Modules and Nichols Algebras

• Classification: All simple Yetter–Drinfeld modules over $H_8$ are classified: eight 1-dimensional and fourteen 2-dimensional simple modules (Shi, 2016).
• Nichols Algebras: The only finite-dimensional Nichols algebras are diagonal (Cartan) types: $A_1$, $A_2$, $A_2 \times A_2$, $A_1 \times \cdots\times A_1$, and $A_1^{\times r}\times A_2$.
• Gelfand–Kirillov Dimension: Used to distinguish finite- from infinite-dimensional Nichols algebras, with a key additivity property under decompositions.

5.2 New Finite-Dimensional Hopf Algebras

By lifting Nichols algebras (bosonization), five new families of finite-dimensional Hopf algebras over $H_8$ are constructed (Shi, 2016); these advance the classification of non-pointed semisimple Hopf algebras outside the group algebra paradigm.

5.3 Simple Partial Comodules

All finite-dimensional simple partial $H_8$-comodules are described using subcentral idempotents in right coideal subalgebras; the partial comodules are classified by this data, with precise simplicity and isomorphism conditions (Batista et al., 2023).

6. Interactions with Quantum Symmetry, Categorification, and Quantum Subgroups

6.1 Quantum Subgroup of $SU_{-1}(2)$

$H_8$ realized as a quantum subgroup via a quotient of $C(SU_{-1}(2))$ (Kitagawa, 2019):

• Graded Twisting: $C(SU_{-1}(2))\cong C(SU(2))^{t,\alpha}$, with $H_8$ arising from restriction to a subgroup $V\subset SU(2)$,
• Fusion Category: The corepresentation (or module) category equates to a Tambara–Yamagami category for $K_4$.

6.2 Symmetry and Entanglement in Field Theory

In 1+1d conformal field theory, the boundary tube algebra (arising in boundary conditions with noninvertible fusion symmetry) in the critical double Ising model is $H_8$ (Choi et al., 4 Sep 2024):

• Symmetry-Resolved Entanglement Entropy: Quantum dimensions and symmetry inherent in $H_8$ control subleading corrections in regimes dominated by noninvertible (“fusion category”) symmetry.

6.3 Bicrossed Products, Matched Pair Actions, and Yang–Baxter Operators

$H_8$ admits matched pair actions, both arising from coquasitriangular structures and distinguished types, yielding six matched pairs in total (Xiao et al., 23 Jan 2025). Exactly two matched pairs induce involutive Yang–Baxter operators, exemplifying set-theoretical solutions to the Yang–Baxter equation in non-pointed, noncommutative settings.

6.4 Invariant 2-Cohomology and Drinfeld Twists

The second invariant cohomology (in both the unitary and invertible settings) for the dual Kac–Paljutkin algebra is trivial (Goswami et al., 9 Oct 2025):

• Categorical Implication: All monoidal autoequivalences of the corepresentation category naturally isomorphic to the identity functor are cohomologically trivial; $H_8$ admits no nontrivial (invariant) Drinfeld twists.

7. Hochschild Cohomology, Smash Products, and PBW Deformations

Smash products involving $H_8$ (and its generalizations $H_{2n^2}$) reveal new structural and homological phenomena:

• Hochschild Cohomology: For any $A$ and semisimple Hopf algebra $H$ (including $H_8$),

$${\bf HH}^\bullet(A\# H) \cong {\bf H}^\bullet(A, A\# H)^H,$$

with explicit chain maps identifying cup product structures (Liu et al., 4 Feb 2025).

• PBW Deformations: All PBW-deformations of $A\sharp H_{2n^2}$ and its Koszul dual are classified, contingent on the module algebra structure of $A$ and its dual. When considering braided smash products (using Drinfeld twist braidings), new PBW-deformations of the corresponding braided tensor product algebra arise, emphasizing the rich deformation theory in the presence of non-group Hopf symmetry (Gao et al., 29 Aug 2024).

---

Through detailed structural, cohomological, and categorical analysis, the Kac–Paljutkin algebra and its generalizations serve as a central testing ground for non-group-type phenomena in Hopf algebras, fusion categories, quantum symmetries, and categorical invariants. This ongoing research continues to illuminate the rigidity, representational richness, and implications for invariant theory, quantum geometry, and categorical representation theory.