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A quiver potential trace is additive on exact sequences

Statement

Let $W$ be a potential of a finite quiver $Q$ , and let $0 \to M_{a} \to M_{a + b} \to M_{b} \to 0$ be represented by a point of the exact-sequence stack. Then $W_{a + b} (M_{a + b}) = W_{a} (M_{a}) + W_{b} (M_{b}) .$

Facts & Assumptions

Given: the block-upper-triangular representation convention of Representation stacks and exact-sequence correspondences of a finite quiver and the cycle-trace definition in Quiver potentials and realized monodromic coefficient data.

[F1]

Representation stacks and exact-sequence correspondences of a finite quiver: every arrow of an extension representation is block upper triangular, with diagonal blocks the subrepresentation and quotient representations.

[F2]

Quiver potentials and realized monodromic coefficient data: a potential is a finite linear combination of oriented cycles modulo cyclic rotation, and $W_{d}$ is the trace of those cycles.

Proof

technique · direct

1.1F1algebra

For a cycle $h_{1}, \dots, h_{ℓ}$ , the product $X_{h_{ℓ}} \dots X_{h_{1}}$ is block upper triangular because products of block-upper-triangular matrices remain block upper triangular. Its two diagonal blocks are respectively the cycle products on $M_{a}$ and $M_{b}$ : this follows inductively from $diag (A B) = diag (A) diag (B)$ for two block-upper-triangular matrices $A, B$ .

2.1step 1.1algebra

The trace of a block-upper-triangular matrix is the sum of the traces of its diagonal blocks. Therefore the trace of this cycle on $M_{a + b}$ is the sum of its traces on $M_{a}$ and $M_{b}$ .

3.1F2step 2.1algebra∎

Write $W$ as a finite sum of cycles with coefficients. Applying step 2.1 to each cycle and summing with those coefficients proves the displayed identity.

Depends on

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Sources

Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §4.7 (standard reference, not scraped)
Ben Davison, The critical CoHA of a quiver with potential, §3.1 (standard reference, not scraped)