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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A two-dimensional comodule and its coefficient matrix

Example

Let $C = E_{2}$ be the $2 \times 2$ (four-dimensional) comatrix coalgebra from Finite comatrix coalgebras exist, with basis $c_{11}, c_{12}, c_{21}, c_{22}$ and structure $Δ (c_{i j}) = \sum_{ℓ = 1}^{2} c_{i ℓ} \otimes c_{ℓ j}, ε (c_{i j}) = δ_{i j} .$ Let $M$ have basis $v_{1}, v_{2}$ and set $ρ (v_{j}) = \sum_{i = 1}^{2} v_{i} \otimes c_{i j} .$ Then $M$ is a right $C$ -comodule and its coefficient matrix is $(c_{i j})$ .

Facts & Assumptions

Given: A field $k$ , the $2 \times 2$ (four-dimensional) comatrix coalgebra structure above, and the two-dimensional vector space $M$ with basis $v_{1}, v_{2}$ .

[F1]

A right coaction must satisfy $(ρ \otimes id) ρ = (id \otimes Δ) ρ$ and $(id \otimes ε) ρ = id$ . (Right comodules, comodule maps, and subcomodules)

[F2]

For a finite basis, coefficient identities $Δ (c_{i j}) = \sum_{ℓ} c_{i ℓ} \otimes c_{ℓ j}$ and $ε (c_{i j}) = δ_{i j}$ define a right-comodule structure by the displayed basis formula. (Kernels, images, quotients, and matrix coefficients of comodules)

[F3]

For $n = 2$ , Finite comatrix coalgebras exist supplies four basis vectors $c_{i j}$ and the displayed coproduct and counit, with a proof that the assignments extend linearly and satisfy coassociativity and both counit identities.

[F4]

A vector space with a finite basis of $n$ elements has dimension $n$ . Thus $M$ has dimension $2$ and $E_{2}$ has dimension $4$ . (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis)

Verification

Given: The basis and coefficient maps of $E_{2}$ and the two-dimensional vector space $M$ .

1.1givenF1construct

Define $ρ$ on $v_{1}, v_{2}$ by the two displayed finite sums and extend linearly. Since $v_{1}, v_{2}$ is a basis, this determines a unique linear map $M \to M \otimes C$ .

2.1F1F3step 1.1algebra

For $j = 1, 2$ , expand the left coaction composite as $(ρ \otimes id) ρ (v_{j}) = \sum_{i, ℓ = 1}^{2} v_{i} \otimes c_{i ℓ} \otimes c_{ℓ j}$ . The right composite is $(id \otimes Δ) ρ (v_{j}) = \sum_{i, ℓ = 1}^{2} v_{i} \otimes c_{i ℓ} \otimes c_{ℓ j}$ using the coproduct formula in [F3]. Thus the composites agree on the basis and hence on every vector.

2.2F1F3step 1.1algebra

For $j = 1, 2$ , $(id \otimes ε) ρ (v_{j}) = \sum_{i = 1}^{2} v_{i} δ_{i j} = v_{j}$ . Linearity extends the counit identity to all of $M$ . Together with the coalgebra structure on $C$ from [F3], this makes the displayed map a right coaction.

3.1F2step 1.1step 2.1step 2.2

By the ordered basis formula for a right coaction, the coefficient in row $i$ , column $j$ is exactly $c_{i j}$ ; the identities in [F2] are the corresponding comatrix identities.

4.1F3F4step 1.1step 2.1step 2.2step 3.1∎

Steps 1.1, 2.1, and 2.2 verify the right coaction equations using the finite comatrix coalgebra of [F3], and step 3.1 identifies its coefficient matrix; [F4] gives the stated dimensions of the two objects.

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Sources

Daniel Kongsgaard and UCSD Math 207A, Hopf Algebras Lecture Notes, Spring 2020 (standard reference, not scraped)
William Chin, A Brief Introduction to Coalgebra Representation Theory (standard reference, not scraped)