Conventions

Axioms

A weak Hopf algebra (Böhm–Nill–Szlachányi) over a field KK is a tuple (A,m,1,Δ,ε,S)(A, m, 1, \Delta, \varepsilon, S) satisfying the axioms below.

Sweedler notation Δ(a)=a(1)⊗a(2)\Delta(a) = a_{(1)} \otimes a_{(2)} is used throughout, with summation implicit.

Hover a property to see which axioms it constrains.

Algebra
(ab)c=a(bc)1a=a1=a\begin{gathered} (ab)c = a(bc) \\ 1a = a1 = a \end{gathered}

Unchanged from the Hopf case. The weakening never touches the algebra structure itself.

Coalgebra
(Δ⊗id)Δ=(id⊗Δ)Δ(ε⊗id)Δ=id=(id⊗ε)Δ\begin{gathered} (\Delta \otimes \mathrm{id})\Delta = (\mathrm{id} \otimes \Delta)\Delta \\ (\varepsilon \otimes \mathrm{id})\Delta = \mathrm{id} = (\mathrm{id} \otimes \varepsilon)\Delta \end{gathered}

Also unchanged. Everything “weak” lives in how the coalgebra interacts with the algebra, not in either one alone.

Multiplicativity of Δ
Δ(ab)=Δ(a) Δ(b)\Delta(ab) = \Delta(a)\,\Delta(b)

Δ is an algebra map. This axiom is identical in the Hopf and weak Hopf settings.

Weak counit
ε(abc)=ε(ab(1)) ε(b(2)c)=ε(ab(2)) ε(b(1)c)\begin{gathered} \varepsilon(abc) = \varepsilon(ab_{(1)})\,\varepsilon(b_{(2)}c) \\ = \varepsilon(ab_{(2)})\,\varepsilon(b_{(1)}c) \end{gathered}

Relaxation of multiplicativity of ε. It becomes ε(ab) = ε(a)ε(b) exactly when Δ(1) = 1 ⊗ 1.

Weak unit
(Δ⊗id)Δ(1)=(Δ(1)⊗1)(1⊗Δ(1))=(1⊗Δ(1))(Δ(1)⊗1)\begin{gathered} (\Delta \otimes \mathrm{id})\Delta(1) \\ = \bigl(\Delta(1) \otimes 1\bigr)\bigl(1 \otimes \Delta(1)\bigr) \\ = \bigl(1 \otimes \Delta(1)\bigr)\bigl(\Delta(1) \otimes 1\bigr) \end{gathered}

Relaxation of Δ(1) = 1 ⊗ 1. This is the axiom whose degeneracy separates weak Hopf from Hopf.

Counital maps
εt(a)=ε(1(1)a) 1(2)εs(a)=1(1) ε(a 1(2))\begin{gathered} \varepsilon_t(a) = \varepsilon(1_{(1)}a)\,1_{(2)} \\ \varepsilon_s(a) = 1_{(1)}\,\varepsilon(a\,1_{(2)}) \end{gathered}

Target and source maps. Their images are the counital subalgebras A_t and A_s.

Antipode
a(1)S(a(2))=εt(a)S(a(1))a(2)=εs(a)S(a(1))a(2)S(a(3))=S(a)\begin{gathered} a_{(1)}S(a_{(2)}) = \varepsilon_t(a) \\ S(a_{(1)})a_{(2)} = \varepsilon_s(a) \\ S(a_{(1)})a_{(2)}S(a_{(3)}) = S(a) \end{gathered}

The three BNS antipode axioms. They reduce to the classical identity when A_t = K1.

Quasitriangularity
Δop(x)R=R Δ(x)(Δ⊗id)R=R13R23(id⊗Δ)R=R13R12\begin{gathered} \Delta^{\mathrm{op}}(x)R = R\,\Delta(x) \\ (\Delta \otimes \mathrm{id})R = R_{13}R_{23} \\ (\mathrm{id} \otimes \Delta)R = R_{13}R_{12} \end{gathered}

Together with the corner axiom Δᵒᵖ(1) R Δ(1) = R and weak invertibility of R.

The single governing invariant of the whole database:

A is a genuine Hopf algebra  ⟺  Δ(1)=1⊗1  ⟺  dim⁡At=1A \text{ is a genuine Hopf algebra} \iff \Delta(1) = 1 \otimes 1 \iff \dim A_t = 1

Hence Hopf⊊weak Hopf\text{Hopf} \subsetneq \text{weak Hopf} is strict: every Hopf algebra is weak Hopf, and any entry with dim⁡At>1\dim A_t > 1 is provably not Hopf.

Fields

Dimension
D=dim⁡KAD = \dim_K A, the dimension of the algebra over the base field.
For A=KGA = K^{\mathcal G}, DD is the number of morphisms of G\mathcal G.
Groupoid label
Every function algebra is KGK^{\mathcal G} for a finite groupoid G\mathcal G.
G\mathcal G is a disjoint union of connected pieces Pairn×H\mathrm{Pair}_n \times H, each with n2∣H∣n^2|H| morphisms.
c x[n·.o·.g·]: c copies, n objects, o = ∣H∣|H|, g = index of HH among groups of order o.
Pieces are joined by +, and D=∑c n2oD = \sum c\,n^2 o.
1x[n1.o4.g2] is C2×C2C_2 \times C_2; 1x[n2.o1.g1] is Pair2\mathrm{Pair}_2; 4x[n1.o1.g1] is 1⊔4\mathbf{1}^{\sqcup 4}.
The table shows the rendered form, with vertex groups named by their GAP structure description.
dim⁡At\dim A_t, dim⁡As\dim A_s
Ranks of the counital maps εt\varepsilon_t, εs\varepsilon_s; their images are AtA_t, AsA_s.
For A=KGA = K^{\mathcal G}: dim⁡At=#Ob(G)\dim A_t = \#\mathrm{Ob}(\mathcal G).
Genuine Hopf
true iff Δ(1)=1⊗1\Delta(1) = 1 \otimes 1, equivalently dim⁡At=1\dim A_t = 1.
An exact zero test over KK. No tolerance is involved anywhere.
Rep
Rep(H)\mathrm{Rep}(H) uses the truncated product Vi⊠Vj=Δ(1)(Vi⊗Vj)V_i \boxtimes V_j = \Delta(1)(V_i \otimes V_j).
The monoidal unit is AtA_t, acting by x▹z=εt(xz)x \triangleright z = \varepsilon_t(xz).
fusion iff that unit is simple; otherwise multifusion.
unknown when the base field does not split the algebra.
QT
true iff a universal R-matrix R∈A⊗AR \in A \otimes A exists.
Corner and intertwining axioms are linear in RR; the two hexagons are quadratic.
Every positive answer is certified against all five axioms, and RR is stored.
Comm, Cocomm
ab=baab = ba for all a,ba, b; and τ∘Δ=Δ\tau \circ \Delta = \Delta with τ\tau the flip.
Both are exact identities on the structure constants.
Semisimple
AA is semisimple as a KK-algebra.
Profile
Artin–Wedderburn profile (n1≥⋯≥nr)(n_1 \ge \dots \ge n_r), ∑ni2=D\sum n_i^2 = D, written n1-n2-….
Every entry is 1-1-…-1: the catalog is the split commutative cell, A≅KDA \cong K^D.
It carries no information beyond DD, so it is omitted from the table.

Scope

Three families
Function algebras KGK^{\mathcal G}: exhaustive, complete for every dimension shown.
Groupoid algebras K[G]K[\mathcal G]: the dual of each entry above, so equally exhaustive.
Named examples: a curated list, not an enumeration. No completeness claim.
Base field
Entries live over Q\mathbb{Q} or a cyclotomic field Q(ζn)\mathbb{Q}(\zeta_n).
Taft algebras need a root of unity, so they cannot live over Q\mathbb{Q}.
An algebra defined over Q\mathbb{Q} may not split there; then REP is unknown.
H8H_8 is stored twice: unknown over Q\mathbb{Q}, fusion over Q(i)\mathbb{Q}(i) with simples 1,1,1,1,21,1,1,1,2.
Not included
Search cells are indexed by Artin–Wedderburn profile, so they presuppose semisimplicity.
Non-semisimple algebras such as H4H_4 and the Taft family are invisible to the search by construction.
Only the commutative cell is complete; cells with a block ni≥2n_i \ge 2 are covered by ansätze only.
Commutative here means split, A≅KDA \cong K^D; Q[C3]≅Q×Q(ζ3)\mathbb{Q}[C_3] \cong \mathbb{Q} \times \mathbb{Q}(\zeta_3) lies outside.
Deduplication
Theorem-driven construction: none needed. Distinct multisets of pieces are non-isomorphic groupoids.
Exhaustive search: SDS_D canonical form, the lexicographic minimum over all D!D! relabelings.
Split commutative: certified by groupoid recovery, which is a complete invariant.
Multimatrix search: an invariant fingerprint. Basis-dependent, hence not an isomorphism invariant.
Distinct cells need no cross-comparison.

Structure constants

Indexing
Bases are indexed 1,…,D1, \dots, D, matching the Julia carrier.
Over Q\mathbb{Q} a coefficient is p or p/q.
Over Q(ζn)\mathbb{Q}(\zeta_n) it is the coordinate vector in 1,ζ,ζ2,…1, \zeta, \zeta^2, \ldots: 1,-1/2,3 is 1−12ζ+3ζ21 - \tfrac{1}{2}\zeta + 3\zeta^2.
RING reads QQ or CYCn.
Tensors
eiej=∑kmult[i,j,k] ek,Δ(ei)=∑j,kcomult[i,j,k] ej⊗eke_i e_j = \sum_k \mathrm{mult}[i,j,k]\, e_k, \qquad \Delta(e_i) = \sum_{j,k} \mathrm{comult}[i,j,k]\, e_j \otimes e_k
1=∑iunit[i] ei,S(ei)=∑jantipode[i,j] ej,R=∑i,jrmatrix[i,j] ei⊗ej1 = \sum_i \mathrm{unit}[i]\, e_i, \qquad S(e_i) = \sum_j \mathrm{antipode}[i,j]\, e_j, \qquad R = \sum_{i,j} \mathrm{rmatrix}[i,j]\, e_i \otimes e_j