Zappa-Szép-Takeuchi Bicrossed Product

• Zappa-Szép-Takeuchi bicrossed product is a construction that intertwines two algebraic structures through mutually compatible actions, ensuring unique factorization.
• It generalizes the matched pair formalism from groups to monoids, semigroups, categories, fusion rings, and Hopf algebras, extending its application to operator algebras and quantum groups.
• The framework underpins key developments in structure theory, automorphism analysis, and homology computations, with implications for exact factorizations and C*-algebra constructions.

The Zappa-Szép-Takeuchi bicrossed product is a categorical and algebraic construction that generalizes the group-theoretical notion of the Zappa-Szép product via matched pairs of actions, extending it to monoids, semigroups, categories, fusion rings, Hopf algebras, and quantum groups. The characteristic feature is the combination of two algebraic structures via mutually compatible actions producing an object whose elements admit unique factorization, intertwining the structures through specified compatibility axioms. In the Hopf algebraic context, this construction is often attributed to Takeuchi; hence, the term bicrossed product or Zappa-Szép-Takeuchi bicrossed product is standard across algebraic literature and operator algebra theory.

1. Fundamental Definitions and Matched Pair Formalism

Let $H$ and $K$ be two groups. A matched pair consists of a left action $\triangleright: K \times H \rightarrow H$ and a right action $\triangleleft: K \times H \rightarrow K$ satisfying the unit and compatibility axioms: $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$ The multiplicative structure is given on $H \times K$ by

$$(h,k)\cdot (h',k') = (h(k\triangleright h'), (k\triangleleft h')k'),$$

with inverse

$$(h,k)^{-1} = (k^{-1}\triangleright h^{-1}, k^{-1}\triangleleft h^{-1}).$$

This yields the bicrossed product $H \bowtie K$, which is a group if and only if the above axioms are satisfied (Lal et al., 2022, Lal et al., 2021, Childs, 2019). If $G$ is a group with subgroups $K$0 satisfying $K$1 and $K$2, then $K$3 is a Zappa-Szép product, and the group structure is captured by a matched pair.

The matched pair formalism generalizes to monoids and semigroups with suitable modifications, e.g., for monoids: $K$4 with corresponding axioms (Loregian, 3 Jan 2025).

2. Algebraic and Categorical Generalizations

The Zappa-Szép-Takeuchi bicrossed product admits generalizations:

• Semigroups: Matched pairs of semigroups $K$5 are defined by actions $K$6, $K$7 satisfying eight axioms generalizing associativity, unitality, and distributivity (Brownlowe et al., 2013).
• Monoids: For monoids $K$8, the bicrossed product $K$9 employs compatible left and right actions, subject to pentagon relations and unit laws, resulting in associative multiplication (Loregian, 3 Jan 2025).
• Categories and Groupoids: For small categories $\triangleright: K \times H \rightarrow H$0 with the same object set, matched pair actions $\triangleright: K \times H \rightarrow H$1, $\triangleright: K \times H \rightarrow H$2 obey pentagonal and object-compatibility axioms, and the resulting product category $\triangleright: K \times H \rightarrow H$3 generalizes group-theoretic Zappa-Szép products to the categorical setting (Mundey et al., 2023, Bédos et al., 2017).
• Fusion Rings and Categories: The bicrossed product construction extends to fusion rings and categories with group-graded bases and tensor compatible actions, yielding bicrossed fusion rings/categories that classify exact factorizations (Müller et al., 2024).

3. Bicrossed Products in Quantum Group and Hopf Algebra Theory

Within Hopf algebras, Takeuchi's bicrossed product uses a pair $\triangleright: K \times H \rightarrow H$4, $\triangleright: K \times H \rightarrow H$5 of Hopf algebras with compatible actions and coactions respecting algebraic structures, such that the tensor product $\triangleright: K \times H \rightarrow H$6 becomes a Hopf algebra under a convolution multiplication: $\triangleright: K \times H \rightarrow H$7 where subscripts denote comultiplication (e.g., $\triangleright: K \times H \rightarrow H$8 under $\triangleright: K \times H \rightarrow H$9) (Childs, 2019, Delvaux et al., 2012). The compatibility conditions ensure associativity, coassociativity, and the antipode structure.

In the category of algebraic quantum groups, i.e., regular multiplier Hopf algebras with integrals, Takeuchi's bicrossed product requires additional compatibility—specifically "twist" and "cotwist" maps—ensuring the existence of integrals and modular data on the product algebra (Delvaux et al., 2012).

4. Structure Theory and Universal Properties

A fundamental feature of the bicrossed product is its universal property:

• For monoids (and analogously for groups, semigroups, and categories), given embeddings $\triangleleft: K \times H \rightarrow K$0, $\triangleleft: K \times H \rightarrow K$1 satisfying intertwining relations,

$\triangleleft: K \times H \rightarrow K$2

$\triangleleft: K \times H \rightarrow K$3 is initial among all such cospans of homomorphisms (Loregian, 3 Jan 2025, Bédos et al., 2017).

• In Garside monoids, the internal Zappa-Szép product $\triangleleft: K \times H \rightarrow K$4 is characterized by unique factorization, with actions preserving the atomic and lattice structures, and Garside's fundamental element splits as $\triangleleft: K \times H \rightarrow K$5 (Gebhardt et al., 2014).

5. Automorphism and Homology Theory

The automorphism group $\triangleleft: K \times H \rightarrow K$6 can be presented as matrices with entries in Hom-sets between the factors and compatibility constraints, generalizing classical descriptions for direct and semidirect products. Central automorphisms are explicitly computed in terms of block-matrix components with constraints arising from preservation of the bicrossed product structure (Lal et al., 2022, Lal et al., 2021).

For category-level bicrossed products, the categorical (co)homology can be computed using double complexes, with explicit spectral sequences relating the homology of the constituent categories and the Zappa-Szép product. For $\triangleleft: K \times H \rightarrow K$7-algebraic deformations, 2-cocycles on the bicrossed product yield twisted Toeplitz–Cuntz–Krieger algebras (Mundey et al., 2023).

6. Operator Algebra and Topological Aspects

In operator algebra theory, Zappa-Szép products of semigroups and small categories serve as sources of new classes of $\triangleleft: K \times H \rightarrow K$8-algebras:

• The semigroup $\triangleleft: K \times H \rightarrow K$9-algebra $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$0 for a Zappa-Szép product $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$1 captures both translation and multiplicative/self-similar components, with boundary quotients generalizing Cuntz-Pimsner algebras (Brownlowe et al., 2013). Examples include the Cuntz algebra $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$2 from the 2-adic adding machine and ring $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$3-algebras of number-theoretic origin.
• For small categories, the bicrossed product construction gives new approaches to Exel-Pardo $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$4-algebras, and the associated crossed product $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$5-algebras unify various graph and Pimsner-Katsura realizations (Bédos et al., 2017).

7. Applications and Examples

The bicrossed product framework unifies a range of phenomena:

• Baumslag–Solitar groups, self-similar group actions, and semigroups of the form $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$6 are realized as Zappa-Szép or bicrossed products (Brownlowe et al., 2013).
• Finite group factorizations: The dihedral group $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$7 as $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$8 with flipping action, and $$\begin{aligned} 1_K\triangleright h &= h, \quad k\triangleright 1_H=1_H, \ 1_K\triangleleft h &= 1_K, \quad k\triangleleft 1_H=k, \ k\triangleright(hh') &= (k\triangleright h)((k\triangleleft h)\triangleright h'), \ (kk')\triangleleft h &= (k\triangleleft (k'\triangleright h))(k'\triangleleft h). \end{aligned}$$9 as $H \times K$0 via a nontrivial matched pair (Childs, 2019).
• Fusion categories: All exact factorizations of fusion rings arise from bicrossed products; explicit fusion rules and associators are computed for Tambara–Yamagami and pointed categories (Müller et al., 2024).
• Homological invariants: Matched pair constructions yield exact sequences relating categorical, diagonal, and total homology; applications include generalized odometer actions (Mundey et al., 2023).

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References:

• (Gebhardt et al., 2014) Zappa-Szép products of Garside monoids
• (Brownlowe et al., 2013) Zappa-Szép products of semigroups and their C*-algebras
• (Lal et al., 2022) Central Automorphisms of Zappa-Szép Products
• (Lal et al., 2021) Automorphisms of Zappa-Szép Product
• (Loregian, 3 Jan 2025) Monads and limits in bicategories of circuits
• (Müller et al., 2024) On bicrossed product of fusion categories and exact factorizations
• (Mundey et al., 2023) Homology and twisted C*-algebras for self-similar actions and Zappa-Szép products
• (Bédos et al., 2017) On finitely aligned left cancellative small categories, Zappa-Szép products and Exel-Pardo algebras
• (Childs, 2019) On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Szép products
• (Delvaux et al., 2012) Bicrossproducts of algebraic quantum groups