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Finite-group Hopf algebra and lattice quantum-double sectors — Examples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, postulates, theorems and proofs

None yet.

4 · Examples, experiments and counterexamples

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

The four electric and magnetic sectors of $D (Z_{2})$

Physical scope

Finite algebraic example for $D (Z_{2})$ and the exact square-torus Pauli-string commutation relation; no general braiding, QFT, or empirical claim.

Example

Assume the Axiom of Choice for the stated lattice prerequisite. Let $Z_{2} = {e, s}$ with $s^{2} = e$ . Its finite-group double has exactly four irreducible complex sectors, each one-dimensional: the two conjugacy-class labels (flux labels) $e, s$ paired with the two group characters $χ_{\pm} : Z_{2} \to C^{\times}$ with $χ_{\pm} (e) = 1$ and $χ_{\pm} (s) = \pm 1$ . Here the conjugacy-class label $C$ is the magnetic-flux label and the centralizer-representation class $[π]$ is the electric-charge label; these are names for the algebraic labels. In particular, the nontrivial charge character evaluated on the nontrivial flux is $- 1$ , equal to the commutation sign for one crossing of a closed primal $Z$ string with a closed dual $X$ string on the square $L \times L$ torus for $L \geq 3$ .

Example references

Irreducible sectors of the finite-group double

Crossing strings and local scalar compression

Facts & Assumptions

Given: The group $Z_{2} = {e, s}$ with $s^{2} = e$ , the finite-group double classification, and the exact square-torus crossing-string relation.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. This is inherited from the lattice theorem's stated interface; the finite group representation calculation itself uses only finite choices.

[F1]

Irreducible sectors of the finite-group double: irreducible $D (G)$ representations are indexed by conjugacy classes $C$ and irreducible representations of the centralizer $Z_{G} (c)$ , with dimension $∣ C ∣ \dim π$ .

[F2]

Crossing strings and local scalar compression: under [A1], for the square $L \times L$ torus with $L \geq 3$ , closed primal $Z$ and closed dual $X$ strings have commutator sign $(- 1)^{I}$ , where $I$ is their mod-two edge-crossing number.

[F3]

Toric-code edge spins on closed-surface cellulations: the square-torus coordinate cycles $p_{1}, q_{1}$ are chosen with mod-two intersection number $I (p_{1}, q_{1}) = 1$ .

Verification

1.1givenF1algebra

The group $Z_{2}$ is abelian, so its two conjugacy classes are the singletons ${e}$ and ${s}$ , and each centralizer is $Z_{2}$ . A group representation is determined by $T = π (s)$ with $T^{2} = I$ . The operators $P_{+} = (I + T) / 2$ and $P_{-} = (I - T) / 2$ satisfy $P_{+}^{2} = P_{+}$ , $P_{-}^{2} = P_{-}$ , $P_{+} P_{-} = 0$ , and $P_{+} + P_{-} = I$ by $T^{2} = I$ ; they commute with $T$ , so their ranges are invariant. Irreducibility forces one range to be all of $V$ and the other zero, giving $T = I$ or $T = - I$ . A scalar action on a complex vector space of dimension greater than one leaves every line invariant, so irreducibility forces dimension one. Conversely each of these two scalar actions defines a character because $(\pm 1)^{2} = 1$ . Thus there are exactly the two characters $χ_{+}$ and $χ_{-}$ for each flux label.

1.2givenalgebra

By the definition of $χ_{-}$ , its value on the nontrivial flux is $χ_{-} (s) = - 1$ .

2.1A1F1F2F3step 1.2algebra∎

The coordinate cycles $p_{1}, q_{1}$ specified in [F3] have $I = 1$ . Applying [F2] to their closed strings gives commutation sign $(- 1)^{1} = - 1$ , equal to $χ_{-} (s)$ . With no crossings, $I = 0$ and the string sign is $+ 1 = χ_{-} (e)$ . The claim is only equality of these finite algebraic signs; it asserts no general braid representation or physical realization.