The closed form of the higher-antibracket coefficients

Abstract

For every n >= 2, the coefficients c_1^n, …, c_n^n that express the higher Koszul bracket Phi^(n+1) through the operators rho_1, …, rho_n are given by the closed formula conjectured by M. Manetti and G. Ricciardi (arXiv:1509.09032, Conjecture 2.4), and (-1)^n c_i^n > 0 for every n >= i >= 1, where c_1^1 = -1. The identity is the one of their Theorem 6.4, stated for linear endomorphisms of Q[x].

Definition 1.1 (The operators Phi^(m,i)).

$$\forall m:\mathbb{N},\forall i:\mathbb{N},\forall s:\mathbb{N},\mathrm{phi}\left(m,i,x^s\right)=\mathrm{ite}\left(s=i,\frac{x^{m-i}}{(m-i)!},0\right)$$

Formalization. D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.phi (✓ std3).

Citation. Marco Manetti; Giulia Ricciardi (2016). Universal Lie Formulas for Higher Antibrackets. DOI: 10.3842/SIGMA.2016.053. URL: https://arxiv.org/abs/1509.09032v3.

Commentary.

Phi^(m,i) is the linear endomorphism of Q[x] sending x^i to x^(m-i)/(m-i)! and every other monomial to 0.

Definition 1.2 (The higher Koszul brackets).

$$\forall m:\mathbb{N},\mathrm{koszul}\left(m\right)=\sum _{i=1}^m(-1{)}^{m-i}\cdot \mathrm{phi}\left(m,i\right)$$

Formalization. D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.koszul (✓ std3).

Citation. Marco Manetti; Giulia Ricciardi (2016). Universal Lie Formulas for Higher Antibrackets. DOI: 10.3842/SIGMA.2016.053. URL: https://arxiv.org/abs/1509.09032v3.

Commentary.

Phi^m is the alternating sum of the operators Phi^(m,i) with signs (-1)^(m-i), for i from 1 to m.

Definition 1.3 (The operators rho_k).

$$\forall k:\mathbb{N},\forall \psi :\mathrm{End}\left(\mathbb{Q}[x]\right),\mathrm{rho}\left(k,\psi \right)=(\frac{x^k}{(k)!}-\frac{x^{k+1}\cdot D}{(k+1)!})\circ \psi -\psi \circ (\frac{x\cdot D^{k+1}}{(k+1)!})$$

Formalization. D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.rho (✓ std3).

Citation. Marco Manetti; Giulia Ricciardi (2016). Universal Lie Formulas for Higher Antibrackets. DOI: 10.3842/SIGMA.2016.053. URL: https://arxiv.org/abs/1509.09032v3.

Commentary.

For a linear endomorphism Psi of Q[x], rho_k(Psi) is the composition of (x^k/k! - x^(k+1) D/(k+1)!) with Psi, minus the composition of Psi with x D^(k+1)/(k+1)!, where D is the derivative of polynomials.

Definition 1.4 (The conjectured coefficient).

$$\forall n:\mathbb{N},\forall i:\mathbb{N},\mathrm{formula}\left(n,i\right)=\frac{(-1{)}^n\cdot {\prod }_{j=2}^i\frac{n\cdot (n-1)-(j-1)\cdot (j-2)}{2}}{{\sum }_{h=2}^{n}h\cdot ({\prod }_{j=2}^h\frac{n\cdot (n-1)-(j-1)\cdot (j-2)}{2})\cdot {\prod }_{j=h}^{n-1}\frac{(1-j)\cdot (j+2)}{2}}$$

Formalization. D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.formula (✓ std3).

Citation. Marco Manetti; Giulia Ricciardi (2016). Universal Lie Formulas for Higher Antibrackets. DOI: 10.3842/SIGMA.2016.053. URL: https://arxiv.org/abs/1509.09032v3.

Commentary.

The coefficient printed in the conjecture: (-1)^n times the product over j from 2 to i of (n(n-1) - (j-1)(j-2))/2, divided by the sum over h from 2 to n of h times the same product up to h times the product over j from h to n-1 of (1-j)(j+2)/2. Empty products are 1.

Definition 1.5 (The conjecture).

$$claim\iff ((\forall n:\mathbb{N},2\le n\Rightarrow \forall c:\mathbb{N}\to \mathbb{Q},\mathrm{koszul}\left(n+1\right)=\sum _{i=1}^n{c}_i\cdot {\left({\mathrm{rho}}_1\right)}^{n-i}(\mathrm{rho}\left(i,\mathrm{koszul}\left(1\right)\right))\iff (\forall i:\mathbb{N},((1\le i)\wedge i\le n)\Rightarrow c_i=\mathrm{formula}\left(n,i\right)))\wedge \left((\forall c:\mathbb{N}\to \mathbb{Q},\mathrm{koszul}\left(1+1\right)=\sum _{i=1}^1{c}_i\cdot {\left({\mathrm{rho}}_1\right)}^{1-i}(\mathrm{rho}\left(i,\mathrm{koszul}\left(1\right)\right))\iff (c_1=-1))\wedge \forall n:\mathbb{N},1\le n\Rightarrow \forall c:\mathbb{N}\to \mathbb{Q},(\mathrm{koszul}\left(n+1\right)=\sum _{i=1}^n{c}_i\cdot {\left({\mathrm{rho}}_1\right)}^{n-i}(\mathrm{rho}\left(i,\mathrm{koszul}\left(1\right)\right)))\Rightarrow \forall i:\mathbb{N},((1\le i)\wedge i\le n)\Rightarrow 0<(-1{)}^n\cdot c_i\right))$$

Formalization. D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.claim (✓ std3).

Citation. Marco Manetti; Giulia Ricciardi (2016). Universal Lie Formulas for Higher Antibrackets. DOI: 10.3842/SIGMA.2016.053. URL: https://arxiv.org/abs/1509.09032v3.

Commentary.

For every n >= 2, a sequence c satisfies Phi^(n+1) = c_1 rho_1^n Phi^1 + c_2 rho_1^(n-2) rho_2 Phi^1 + … + c_n rho_n Phi^1 exactly when c_i equals the printed coefficient for 1 <= i <= n. For n = 1 the identity Phi^2 = c_1 rho_1 Phi^1 holds exactly when c_1 = -1. For every n >= 1, every solution c satisfies (-1)^n c_i > 0 for 1 <= i <= n. Here rho_1^(n-i) applies rho_1 n - i times. The paper proves that this identity has a unique solution and that the coefficients of its Theorem 2.3 are that solution, so the three parts are the formula of the conjecture for n >= 2 and its sign clause for n >= i >= 1.

Theorem 1.6 (The closed form holds).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.result (✓ std3). ∎

Resolves. Problems/manetti-ricciardi-2015-higher-antibracket-coefficients (proved) by D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.result.

Source. Repository-derived.

Acknowledgement. Marco Manetti; Giulia Ricciardi (2016). Universal Lie Formulas for Higher Antibrackets. DOI: 10.3842/SIGMA.2016.053. URL: https://arxiv.org/abs/1509.09032v3.

Commentary.

Write an operator through its values on monomials and record the coefficient of x^d/d! in its value at x^s as the lattice point (d, s). Then rho_k sends (d, s) to alpha_k(d) (d + k, s) - C(s + k, k + 1) (d, s + k) with alpha_k(d) = C(d + k, k) - C(d + k, k + 1), and rho_i Phi^1 = (i, 1) - (0, i + 1). The two moves of rho_1 commute, so rho_1^t expands binomially along products of the step weights. The identity becomes n + 1 linear equations in c, one for each lattice point (d, n + 1 - d). Since alpha_1(2) = 0, the equations for d >= 3 form a triangular system in c_3, …, c_n, and those for d = 0 and d = 1 then fix c_1 and c_2, which gives uniqueness. For c_i = (-1)^n (n+i-2)! / ((n-i)! 2^(i-1)) divided by (n-2)! (n+1)! / 2^(n-1), every equation reduces to the alternating binomial sum over i of (-1)^(M-i) C(M, i) C(i + a, b), which is C(a, b - M) for M <= b and 0 otherwise. The same sum shows that the printed product and denominator equal these factorial expressions for n >= 3; the case n = 2 is computed directly and gives c_1 = c_2 = 1/2, and for n = 1 the two equations give c_1 = -1.

References

• Truth anchor: D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.claim
• Truth anchor: D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.formula
• Truth anchor: D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.koszul
• Truth anchor: D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.phi
• Truth anchor: D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.result
• Truth anchor: D5/S3/HomologicalAlgebra/HigherAntibracketCoefficients.rho