bibkey: manetti2016antibrackets authors: Marco Manetti; Giulia Ricciardi year: 2016 title: “Universal Lie Formulas for Higher Antibrackets” doi: 10.3842/SIGMA.2016.053 url: https://arxiv.org/abs/1509.09032v3 claim: “In the algebra of linear endomorphisms of Q[x] vanishing at 1, with Phi^{n,i}(x^i) = x^{n-i}/(n-i)!, Phi^n = sum_i (-1)^{n-i} Phi^{n,i} and rho_k(Psi) = (x^k/k! - x^{k+1} d/(k+1)!) o Psi - Psi o x d^{k+1}/(k+1)!, there is for every n > 0 a unique sequence of rationals c_1^n, …, c_n^n with Phi^{n+1} = (c_1^n rho_1^n + c_2^n rho_1^{n-2} rho_2 + … + c_n^n rho_n) Phi^1 (Theorem 6.4); these are the coefficients of Theorem 2.3 for higher Koszul brackets. Conjecture 2.4 gives a closed formula for c_i^n, verified by the authors for n <= 12, and asserts (-1)^n c_i^n > 0.” strata_touched:

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Universal Lie Formulas for Higher Antibrackets

Marco Manetti, Giulia Ricciardi, SIGMA 12 (2016), 053, 20 pages; arXiv:1509.09032 (math.QA, cross-listed hep-th and math-ph). Quotations are from the arXiv v3 source.

Theorem 2.3:

In the notation above, for every integer $n>0$, there exists an unique sequence $c_1^n,\dots ,c_n^n$ of rational numbers such that, for every linear operator $f:A\to A$, we have

followed by the display ${\Phi }_f^{n+1}=(c_1^n{\rho }_1^n+c_2^n{\rho }_1^{n-2}{\rho }_2+c_3^n{\rho }_1^{n-3}{\rho }_3+\cdots +c_n^n{\rho }_n)f$.

Conjecture 2.4:

For every $n\ge 2$ the coefficients $c_i^n$ of Theorem~{\rm \ref{thm.standardform}} are given by the formula

followed by the display $c_i^n=\frac{(-1{)}^n\prod _{j=2}^i\frac{n(n-1)-(j-1)(j-2)}{2}}{\sum _{h=2}^{n}h\left(\prod _{j=2}^h\frac{n(n-1)-(j-1)(j-2)}{2}\right)\left(\prod _{j=h}^{n-1}\frac{(1-j)(j+2)}{2}\right)}$, and

where every empty product is intended to be equal to $1$. Moreover $(-1{)}^n{c}_i^n>0$ for every $n\ge i\ge 1$.

The operators (Section 6):

For every $n\ge i>0$, denote by ${\Phi }^{n,i}\in \mathfrak{a}$ the operator

with ${\Phi }^{n,i}(x^i)=\frac{x^{n-i}}{(n-i)!}$ and ${\Phi }^{n,i}(x^s)=0$ for $s\ne i$; ${\Phi }^n={\sum }_{i=1}^n(-1{)}^{n-i}{\Phi }^{n,i}$; and Lemma 6.2 defines $\rho (l_n)(\Psi )=\left(\frac{x^n}{n!}-\frac{x^{n+1}\partial }{(n+1)!}\right)\circ \Psi -\Psi \circ \frac{x{\partial }^{n+1}}{(n+1)!}$. Theorem 6.4 states the unique solvability of ${\Phi }^{n+1}=(c_1^n{\rho }_1^n+c_2^n{\rho }_1^{n-2}{\rho }_2+\cdots +c_n^n{\rho }_n){\Phi }^1$ in $\mathfrak{a}$, and Lemma 6.3 transports it to Theorem 2.3.

Verified locator

• DOI: https://doi.org/10.3842/SIGMA.2016.053 (SIGMA 12 (2016), 053).
• URL: https://arxiv.org/abs/1509.09032v3 (v1 2015-09-30, v2 2015-11-12, v3 2016-06-06; source sigma16-053.tex, md5 69e1f0c5a2a7c94f98ac2352869f2b60): Theorem 2.3 (l. 176–181), the sentence before the conjecture (l. 184), Conjecture 2.4 (l. 186–192), the algebra $\mathfrak{a}$ (l. 662), ${\Phi }^{n,i}$ (l. 681–682), ${\Phi }^n$ (l. 686–688), Lemma 6.2 (l. 700–703), Lemma 6.3 (l. 735–749) and Theorem 6.4 (l. 751–754).