Splitting of the Markov CoHA into spherical and exterior parts

Determine whether the cohomological Hall algebra Coha_{Q,W} for the Markov quiver Q with the homogeneous potential W = c1 b1 a1 + c2 b2 a2 decomposes, as a graded algebra, as the direct product of the spherical subalgebra S_Q and the free exterior algebra A generated by the anti-invariant classes under the involution swapping the arrow pairs (a1,a2), (b1,b2), and (c1,c2).

Background

In Section 3.2, the paper analyzes the non-spherical part of the cohomological Hall algebra (CoHA) for the Markov quiver with the homogeneous cubic potential W = c1 b1 a1 + c2 b2 a2. A Z/2Z symmetry swapping the arrow pairs acts trivially on the spherical subalgebra but nontrivially on certain BPS classes, yielding an anti-invariant summand.

The authors identify specific generators for the non-spherical component and note that the algebra generated by these elements surjects onto a free exterior algebra A. Motivated by physics, they propose that the full CoHA might split as the product of this exterior algebra A and the spherical subalgebra S_Q, and suggest that this prediction can be tested for low dimension vectors via dimensional reduction and Proposition 3.4.