面向自由概率的非交换多元幂级数移位替换
Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
AI总结：
本研究探讨非交换变量形式幂级数上由分次连通字Hopf代数线性形式解释诱导的左线性群运算律及其关联预李结构，证明其可群论化表述自由概率等非交换概率理论中的核心恒等式与变换。
摘要：
我们研究非交换变量形式幂级数上的一种特定群运算律，该运算律由幂级数作为合适的分次连通字Hopf代数上线性形式的解释诱导而来。该群运算律是左线性的，因此与形式幂级数上的一个预李结构相关联。我们研究了这些结构，并展示了如何利用它们以群论形式重新表述形式幂级数上的各类恒等式与变换，这些内容在非交换概率论，尤其是Voiculescu的自由概率理论中一直处于核心地位。
英文摘要：
We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.
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