From Lie--Rinehart Algebras to $F$-Manifold Algebras

Abstract: For every Lie--Rinehart algebra, we construct an $F$-manifold algebra on the direct sum of its base algebra and module. Contrary to the assertion in \cite[Proposition 13.3.26]{LodayVallette}, the resulting structure is generally not Poisson. We determine when powers of the positive-degree ideal in the associated symmetric Poisson algebra are Poisson ideals, and relate the Leibnizator to the Lie--Rinehart differential. For a finite projective module of constant rank, the trace of the Leibnizator recovers the anchor and yields a rigidity result for injective anchors. We conclude with algebraic and geometric examples.

• The paper constructs an F-manifold algebra on A⊕L from every characteristic-zero Lie–Rinehart algebra, with the Leibnizator measuring the anchor’s failure to satisfy the Poisson derivation rule.
• It proves that the truncation Sym_A(L)/I^n is Poisson exactly when the anchor’s induced contraction vanishes, and for finite projective L of positive constant rank this occurs only when the anchor is zero.
• For finite projective L of constant rank r, the anchor is recovered from the Leibnizator by ρ(X)(a)=(r+1)⁻¹Tr(Y↦⟨_a(X,Y)), showing that faithful anchors determine the full Lie–Rinehart structure and correcting an unconditional Poisson claim.

Overview and main construction

This paper establishes a direct functorial relationship between Lie–Rinehart algebras and $F$-manifold algebras. Given a Lie–Rinehart algebra $(A,L,[-,-]_L,\rho)$ over a field of characteristic zero, the authors equip the direct sum $P=A\oplus L$ with the commutative associative product

$(a+X)\bullet(b+Y)=ab+aY+bX$

and the semidirect Lie bracket determined by the anchor and the bracket on $L$:

$[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$

The central computation is the Leibnizator of this structure:

$_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$

Because the Leibnizator does not vanish in general, $(A\oplus L,\bullet,[-,-])$ is not a Poisson algebra; nevertheless, the derivation property of the anchor implies that it satisfies the Hertling–Manin identity, so it is an $F$-manifold algebra (2608.12802). The proof reduces cleanly: since elements act on $L$ through their $(A,L,[-,-]_L,\rho)$0-component only, the Hertling–Manin identity follows from the Leibniz rule for the Lie–Rinehart differential $(A,L,[-,-]_L,\rho)$1, $(A,L,[-,-]_L,\rho)$2.

Correction to a claim in Loday–Vallette

A notable point is that the paper contradicts Proposition 13.3.26 of Algebraic Operads by Loday and Vallette, which asserts that $(A,L,[-,-]_L,\rho)$3 with these operations is a Poisson algebra for an arbitrary Lie–Rinehart algebra. The Leibnizator formula shows this holds only under the additional condition $(A,L,[-,-]_L,\rho)$4 for all $(A,L,[-,-]_L,\rho)$5. This is a substantive correction: the correct unconditional conclusion is the weaker $(A,L,[-,-]_L,\rho)$6-manifold algebra statement.

Poisson truncations of the symmetric algebra

As a commutative algebra, $(A,L,[-,-]_L,\rho)$7, where $(A,L,[-,-]_L,\rho)$8 is the positive-degree ideal of the canonical linear Poisson algebra $(A,L,[-,-]_L,\rho)$9. The paper determines exactly when the linear Poisson bracket descends to these truncations:

• First-order truncation: $P=A\oplus L$0 is a Poisson ideal if and only if $P=A\oplus L$1 for all $P=A\oplus L$2.
• Higher truncations: writing $P=A\oplus L$3 via the contraction map $P=A\oplus L$4, the ideal $P=A\oplus L$5 is Poisson precisely when $P=A\oplus L$6 for all $P=A\oplus L$7.
• Rigidity criterion: when $P=A\oplus L$8 is finite projective of positive constant rank, $P=A\oplus L$9 is injective (using localization at primes and invertibility of $(a+X)\bullet(b+Y)=ab+aY+bX$0 in characteristic zero), so $(a+X)\bullet(b+Y)=ab+aY+bX$1 is a Poisson ideal for some $(a+X)\bullet(b+Y)=ab+aY+bX$2 — equivalently every $(a+X)\bullet(b+Y)=ab+aY+bX$3 — if and only if $(a+X)\bullet(b+Y)=ab+aY+bX$4.

Consequently, for any nonzero anchor on a finite projective module of positive constant rank, no quotient $(a+X)\bullet(b+Y)=ab+aY+bX$5 inherits the Poisson structure, even though $(a+X)\bullet(b+Y)=ab+aY+bX$6 always holds (the ideal is coisotropic but not Poisson).

Recovery of the anchor from the Leibnizator

The paper's most striking quantitative result concerns the trace of the Leibnizator. For $(a+X)\bullet(b+Y)=ab+aY+bX$7 finite projective of constant rank $(a+X)\bullet(b+Y)=ab+aY+bX$8, define $(a+X)\bullet(b+Y)=ab+aY+bX$9. Then

$L$0

so the anchor is recovered from the Leibnizator by

$L$1

The proof uses the dual basis lemma: the endomorphism $L$2 decomposes into scalar multiplication by $L$3 (trace $L$4) plus a rank-one term (trace $L$5). Two consequences follow:

1. Differential interpretation: the Leibnizator satisfies $L$6, so the Hertling–Manin identity is equivalent to the Leibniz rule $L$7.
2. Rigidity: two Lie–Rinehart structures on the same pair $L$8 with equal Leibnizators have equal anchors; if the common anchor is injective, the brackets coincide as well. Thus the $L$9-manifold algebra remembers the entire Lie–Rinehart structure whenever the anchor is faithful.

The paper also identifies two canonical Poisson subalgebras of $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$0: the invariant part $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$1 (functions constant along the anchor distribution) and the isotropy part $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$2, intersecting in $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$3.

Examples

Three classes illustrate the construction. For a Lie algebroid $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$4, one obtains an $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$5-manifold algebra on $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$6, identified with the degree-at-most-one truncation of fiberwise polynomial functions on $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$7 under the linear Poisson structure. For the polynomial derivation Lie–Rinehart algebra $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$8, the Leibnizator evaluates as $[a+X,b+Y]=\rho(X)(b)-\rho(Y)(a)+[X,Y]_L.$9, giving a concrete witness that the structure is genuinely non-Poisson. In a Witt-type basis $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$0, $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$1, the Leibnizator takes the form $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$2 for $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$3. Finally, for the cotangent Lie–Rinehart algebra of a Poisson algebra, the associated $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$4-manifold algebra on $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$5 recovers the original Poisson tensor via the trace formula when $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$6 is finite projective of constant rank — e.g., for the symplectic affine plane, $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$7 confirms non-Poisson behavior while the trace reconstructs $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$8.

Limitations and open questions

Several restrictions bound the results. The trace-based recovery of the anchor requires $_{a+X}(b+Y,c+Z)=\rho(Y)(a)Z+\rho(Z)(a)Y.$9 finite projective of constant rank; without this hypothesis, $(A\oplus L,\bullet,[-,-])$0 need not be injective and the equivalence between Poisson ideals and vanishing anchor fails. The rigidity corollary likewise depends on injectivity of the anchor, leaving open the case of non-faithful anchors, where distinct Lie–Rinehart brackets can share the same Leibnizator. The paper works throughout over a field of characteristic zero, and the invertibility of $(A\oplus L,\bullet,[-,-])$1 and $(A\oplus L,\bullet,[-,-])$2 used in the proofs does not extend verbatim to positive characteristic or to base rings. The paper also leaves open whether the construction admits a functorial characterization among $(A\oplus L,\bullet,[-,-])$3-manifold algebras, and whether higher-order analogues of the Hertling–Manin identity govern deeper truncations beyond the first-order analysis given here.

Conclusion

The paper provides a uniform passage from Lie–Rinehart algebras to $(A\oplus L,\bullet,[-,-])$4-manifold algebras, corrects the Poisson claim in Loday–Vallette, classifies precisely when powers of the positive-degree ideal are Poisson ideals, and shows that in the finite projective constant-rank setting the Leibnizator encodes both the anchor and — under faithfulness — the full Lie–Rinehart structure. The combination of the trace formula and the rigidity result indicates that the $(A\oplus L,\bullet,[-,-])$5-manifold algebra on $(A\oplus L,\bullet,[-,-])$6 carries substantially more information than its definition suggests.