Injectivity of the universal-enveloping-algebra map

Prove or disprove injectivity of the algebra homomorphism U(ι): U(Gr^Y𝓘𝓒) → Gr^Fℤ[𝓘𝓒] induced by the canonical map from the Lie algebra of homology cylinders to the associated graded Goussarov–Habiro algebra.

Background

The paper constructs a surjective algebra homomorphism from the universal enveloping algebra of the graded Lie algebra of homology cylinders to the associated graded algebra of the monoid algebra under the Goussarov–Habiro filtration.

Surjectivity is established, but the integral injectivity question is left unresolved. The authors contrast this with the situation over a field, where an analogue is an isomorphism under the appropriate interpretation of the universal enveloping algebra.

References

The injectivity of the map $\operatorname{U}(\iota)$ does not seem to be known.

— On the Sp-structure of the torsion of the Lie algebra of homology cylinders  (2609.26409 - Faes et al., 22 Sep 2026) in Remark following Theorem 2.3 in Section 2.2, “The algebra of homology cylinders”