非交换多重指标的平面霍普夫代数
The planar Hopf algebra of noncommutative multi-indices
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中文总结 AI 辅助
构建平面Linares--Otto--Tempelmayr霍普夫代数，填补相关正方形的非交换多重指标空缺。从自由结合代数出发经特定构造得到该代数，引入平面树繁殖力映射并证明其同构性，还推导余积公式等，建立与其他霍普夫代数的联系。
中文摘要 AI 辅助
我们构建了平面Linares--Otto--Tempelmayr霍普夫代数，填补了与LOT、Butcher--Connes--Kreimer和Munthe-Kaas--Wright霍普夫代数相关的正方形中缺失的平面非交换多重指标一角。从加权字母表\(\mathbb Z_{\ge -1}\times A\)上的自由结合代数出发，定义插入型乘积，经Guin--Oudom构造得到平面LOT霍普夫代数。引入从装饰平面有根树到\(V(A)\)中单项式的平面树繁殖力映射，证明其为线性同构，得到与Munthe-Kaas--Wright霍普夫代数的自然同构。还推导了基于左容许切割的显式余积公式，建立提取-收缩余积，并构造了与经典树对称化算子兼容的单词对称化算子。
英文摘要
We construct the planar Linares--Otto--Tempelmayr Hopf algebra, thereby filling the missing planar noncommutative multi-index corner in the square relating the LOT, Butcher--Connes--Kreimer, and Munthe-Kaas--Wright Hopf algebras. Starting from the free associative algebra on a weighted alphabet $\mathbb Z_{\ge -1}\times A$, we define an insertion-type product yielding a post-Lie structure on the Lie algebra generated by the linear span $V(A)$ of weight $-1$ monomials whose proper left prefixes all have nonnegative weight, and the Guin--Oudom construction then produces the planar LOT Hopf algebra. We introduce a planar tree fertility map from decorated planar rooted trees to monomials in $V(A)$, prove that it is a linear isomorphism, and obtain a natural Hopf algebra isomorphism with the Munthe-Kaas--Wright Hopf algebra. We further derive an explicit coproduct formula in terms of left-admissible cuts, establish the extraction-contraction coproduct, and construct a word symmetrization operator compatible with the classical tree symmetrization operator.
