False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-04

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FALSE: the dimension of an object is independent of the pivotal structure

Statement

The dimension of an object is independent of the pivotal structure.

Facts & Assumptions

Given: The tensor category of finite-dimensional $Z$ -graded vector spaces, a field element $q \in k^{\times}$ with $q \neq \pm 1$ , and the degree-one line $X = k [1]$ .

[L1]

Pivotal structures vary by monoidal automorphisms of the identity (Pivotal and spherical structures vary by monoidal automorphisms of the identity).

[L2]

$\dim_{a} (X)$ is defined as the trace of the chosen pivotal comparison (The dimension of an object relative to a pivotal structure).

[L3]

On graded vector spaces, multiplying the canonical double-dual map by $q^{n}$ in degree $n$ gives a pivotal structure whose dimension on $k [1]$ is $q$ (FALSE: the left and right traces always agree).

Refutation

technique · direct

1.1

By [L1], the standard pivotal structure on graded vector spaces can be multiplied by the monoidal automorphism acting as $q^{n}$ in degree $n$ . This gives the modified pivotal structure $a^{'}$ used in [L3].

givenL1L3

2.1

The standard pivotal structure gives $\dim (X) = 1_{k}$ for the degree- $1$ line $X = k [1]$ , while [L3] gives $\dim_{a^{'}} (X) = q \neq 1_{k}$ . These are exactly the quantities defined in [L2].

step 1.1L2L3

3.1

Therefore the dimension can change when the pivotal structure changes, so the statement is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Exercise 4.7.16 (standard reference, not scraped)