Extension of factorisation to higher products

Extend the factorisation of the toy minimal $A_\infty$-algebra from the product $m_2$ and the trilinear product $m_3$ to higher products $m_n$ with $n>3$, including the appropriate higher analogues of the insertion identities.

Background

The paper proves a lift of factorisation to m3m_3 for a toy model whose ideal is generated by a central, pi-invariant element. The construction requires compatibility conditions involving m3m_3, the projection onto representatives, and the map defining the ideal component.

The authors propose that the identities obtained by inserting the ideal generator into all argument slots should generalize to higher products, but the extension beyond m3m_3 is not developed.

References

The extension of the construction to $m_n$ with $n > 3$, where we expect tau-insertion with $n \geqslant 3$ to play the role of commutation-generalisation, is left for future work.

— An explicit product for the on-shell higher-spin algebra in $AdS_d$-space  (2610.26704 - Povarnin, 19 Oct 2026) in Section 5.4, 'Lifting the factorisation to the lowest higher product'; Conclusion