bibkey: burmesterkuhn2025postlie
authors: Annika Burmester; Ulf Kühn
year: 2025
title: "On post-Lie structures for free Lie algebras"
doi: 10.48550/arXiv.2504.19661
url: https://arxiv.org/html/2504.19661v1
claim: "Conjecture 4.29: for every k >= 0 the ari coproduct preserves the increasing k-level filtration in its right tensor factor on all rational word polynomials."
strata_touched:
  - D5/S1/Words/PostLie/AriWordOperations
  - D5/S1/Words/PostLie/AriProjection
  - D5/S1/Words/PostLie/AriWordProduct
  - D5/S1/Words/PostLie/AriCoproductFiltration
license: citation-only
triage: anchor

Ari coproduct and the right-factor level filtration

Verified locator

The original source is Annika Burmester and Ulf Kühn, On post-Lie structures for free Lie algebras, arXiv:2504.19661v1, submitted 28 April 2025. The verified DOI is 10.48550/arXiv.2504.19661; the original full text is https://arxiv.org/html/2504.19661v1, its TeX is https://arxiv.org/src/2504.19661v1, and its version history is https://arxiv.org/abs/2504.19661. The inspected history lists v1 only.

The target is the whole Conjecture 4.29. Its carrier is the finite rational concatenation-word algebra on v_0,v_1,..., including the empty word. For all natural k,r, let lambda_k(w) count letters different from v_k and let F_r^k be the rational span of words with lambda_k(w) <= r. The assertion is

[ \Delta_a(F_r^k)\subseteq \operatorname{image}\bigl(H\otimes_{\mathbb Q}F_r^k \longrightarrow H\otimes_{\mathbb Q}H\bigr). ]

No restriction to positive indices, homogeneous inputs, a fixed word length, or a single generator is part of this assertion.

Original finite operations and pairing

Definition 4.2 specifies beta(0,0)=1, both mixed-zero values zero, and beta(a,l)=choose(a-1,l-1) when a,l>0. The signed multiplicity of equal-length index tuples is

[ \mu_{\mathbf a,\mathbf l} =(-1)^{\sum a_i+\sum l_i}\prod_i\beta(a_i,l_i). ]

For nonzero multiplicity, each l_i <= a_i and l_i=0 exactly when a_i=0. The converse stated in Remark 4.3(i) fails at mixed-zero pairs and is not used.

The letter-action presentation is the first equality in the full proof of Proposition 4.19, taking its Lie factors to be individual generators. Lemma 4.10 derives that equality from the magmatic Definition 4.8 and its extension; Definition 1.9 is the enveloping extension. Thus for a nonempty concatenation word A=v_(a_1)...v_(a_m) and s>0, the original action is the finite sum

[ A\triangleright_a v_s =\sum_{0\le l_i\le a_i}\mu_{\mathbf a,\mathbf l} [\cdots[[v_{s+\sum_i(a_i-l_i)},v_{l_1}],v_{l_2}],\ldots,v_{l_m}], \qquad [p,q]=pq-qp. ]

The empty word acts identically on every generator. Every nonempty word acts by zero on v_0. This is the presentation used here; the different naive associative extension excluded in Remark 4.16 is not used. No additional magmatic or enveloping realization is postulated.

Proposition 2.1 gives A star B by assigning every position of A to one of length(B)+1 slots. Letters in each slot retain their original order. A term is slot zero followed by, in the order of the letters of B, the action of slot j+1 on letter j of B. Every assignment is counted, even if repeated letters give the same resulting tuple. Primitive unshuffle sends a generator to the sum of its placements in the individual slots; its algebra-homomorphism extension gives these assignments with their multiplicities.

The ordinary word pairing is equation (18). Definition 1.20 characterizes the dual coproduct uniquely, and Remark 1.22 describes its coefficients as [A tensor B] Delta_a(w) = [w](A star B). The ordinary pairing is symmetric. Its symmetry identifies the source convention, which pairs the product on the left, with the coefficient convention that pairs the input basis word on the left. There is no factorial or divided-power normalization. Example 4.28 omits the two unit tensor terms in its display; the full defining pairing, rather than a reduced-coproduct convention, specifies the operation here. The source weight is gamma(0)=1 and gamma(i)=i for positive i.

Projection and finite transpose argument

The concatenation algebra map pi_k:H -> Q[t] sends v_k to t and every other generator to zero. For every finite rational polynomial P and every natural d, including zero, its t^d coefficient equals the coefficient of the unique pure word v_k^d in P. Every ordinary commutator projects to zero. Consequently a nonempty left word acting on any generator has zero pure projection. The empty-left-word branch is the original generator. Thus an action output for v_s has at least one letter different from v_k whenever s != k.

Concatenation adds these counts. Finite sums and rational cancellation cannot create support outside the union of the summand supports. Applying this fact to the ordered unshuffle formula gives

[ [w](A\star_a B)\ne0\quad\Longrightarrow\quad \lambda_k(B)\le\lambda_k(w). ]

The nonzero multiplicity condition preserves zero positions and hence the original weight despite the shifted positive index. The action and product are homogeneous for this weight. A word of weight at most N has length at most N and all indices at most N. Bounded lists over the genuinely finite alphabet Fin(N+1), filtered by weight, therefore give exactly the finite window of such words. For a word w of weight N, the tensor sum with coefficients [w](A star B) over that window has the full ordinary transpose coefficients for every pair, including pairs outside the window. It is an ordinary algebraic tensor, not a completed tensor series.

The coefficient support bound places that tensor in the span of pairs whose second word has level at most lambda_k(w). Each spanning tensor is the image of an actual element of H tensor F_r^k. Rational linearity then gives the full assertion for arbitrary P in the rational span. These are classical projection and support ingredients; their use here is a synthesis directed at the specific original assertion.

Bounded prior-art limits

The inspected relevant passages of arXiv:2508.05024v1, 2603.05038v1, 2609.25168v1, and 2610.37351v1 did not identify a settlement of the whole Conjecture 4.29. MathDB page 368177 concerns Appendix A.7’s uri multiplicity threshold shuffle assertion and is a different problem. The bounded source-title, author, and operation searches did not identify an exact posted solution. Rate-limited citation APIs do not establish a complete citation search. These observations assert neither worldwide absence nor priority.