PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Positive weights characterize the specified Hermitian form

Statement

For the semisimple closed model determined by the stated idempotent data, define $a^{*} = \sum_{i} \overline{a_{i}} e_{i}$ when $a = \sum_{i} a_{i} e_{i}$ , and set $h (a, b) = ϵ (a^{*} b)$ . The form $h$ is a Hermitian positive-definite inner product exactly when every weight $w_{i}$ is a positive real number. The algebraic closed oriented TQFT exists for every nonzero complex weight; positivity is an additional condition for this specified Hilbert-space interpretation.

Statement references

Semisimple closed two-dimensional TQFT data

Facts & Assumptions

Given: The semisimple model and its idempotent basis from Semisimple closed two-dimensional TQFT data. The involution is conjugate-linear and fixes each $e_{i}$ . We use the convention that a sesquilinear form is conjugate-linear in its first argument and linear in its second; it is Hermitian when $h (a, b) = \overline{h (b, a)}$ , and positive definite when $h (a, a) > 0$ for every $a \neq 0$ .

[F1]

Semisimple closed two-dimensional TQFT data: $A = ⨁_{i} C e_{i}$ , $e_{i} e_{j} = δ_{i j} e_{i}$ , and $ϵ (e_{i}) = w_{i}$ .

[F2]

Semisimple closed two-dimensional TQFT data: every nonzero complex weight gives the stated algebraic closed oriented TQFT through the closed classification; no positivity hypothesis occurs there.

Proof

1.1givenF1algebra

If $a = \sum_{i} a_{i} e_{i}$ and $b = \sum_{i} b_{i} e_{i}$ , then $a^{*} b = \sum_{i} \overline{a_{i}} b_{i} e_{i}$ , so $h (a, b) = \sum_{i} w_{i} \overline{a_{i}} b_{i}$ .

2.1givenstep 1.1algebra

If every $w_{i}$ is positive real, the formula in step 1.1 is Hermitian and $h (a, a) = \sum_{i} w_{i} ∣ a_{i} ∣^{2} > 0$ for each nonzero $a$ , because at least one coefficient is nonzero.

2.2givenF1step 1.1algebra

Conversely, if $h$ is Hermitian and positive definite, then $w_{i} = h (e_{i}, e_{i})$ is real by Hermitian symmetry and strictly positive by positive definiteness for every nonzero basis vector $e_{i}$ . Thus all weights are positive real, proving the equivalence.

3.1givenF2step 2.2∎

The classification used in [F2] assumes only the nonzero Frobenius weights, so the algebraic closed theory remains defined when the extra positivity condition fails; this separates the categorical model from the specified Hilbert-space interpretation.

Form matrix

For the basis $(e_{i})$ , the associated sesquilinear form has diagonal matrix $diag (w_{1}, \dots, w_{r})$ . Positivity is a property of this explicit matrix and does not follow from the existence of the oriented closed TQFT.

Depends on

Semisimple closed two-dimensional TQFT data