Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability

IF 0.9 3区物理与天体物理 Q2 MATHEMATICS

Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-04-04 DOI:10.3842/SIGMA.2023.038

K. Ebrahimi-Fard, F. Patras, N. Tapia, L. Zambotti

引用次数: 1

Abstract

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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从自由概率的角度看非交换多元幂级数的移位代换

我们研究了非交换变量中形式幂级数的一个特殊群律，该群律是由它们在适当的分次连接词Hopf代数上的线性形式解释引起的。这个群定律是左线性的，因此与形式幂级数上的前李结构有关。我们研究了这些结构，并展示了它们如何被用来以群论的形式重塑形式幂级数上的各种恒等式和变换，这些恒等式和转换在非交换概率论的背景下一直是核心，特别是在Voiculescu的自由概率论中。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Symmetry Integrability and Geometry-Methods and Applications 物理-物理：数学物理

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

4-8 weeks

期刊介绍： Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.