非交换概率论中的代数群再探讨
Algebraic groups in non-commutative probability theory revisited
AI总结：
本文旨在统一非交换概率论中基于泛积和基于Hopf代数的两种代数群方法，利用代数群与shuffle群技术证明了自由、Boolean及单调概率理论中代数群概念的等价性，并给出了Hopf代数结构的显式公式。
摘要：
von Waldenfels和Schürmann学派长期以来一直倡导余代数以及代数群在非交换概率论中的作用。最近引入了另一种代数方法，基于shuffle和pre-Lie演算，并给出了另一个编码状态行为的特征群构造。比较两者，第一种方法最近由Manzel和Schürmann用一般范畴论语言重新表述，可以被视为在很大程度上由泛积理论驱动；而第二种构造则建立在Hopf代数以及对非交叉集合划分组合学的适当代数化基础之上。尽管两者都处理相同的现象，但在两种观点之间进行转换并不显然。我们在此提出一种统一这两种方法的尝试，通过显式揭示它们之间的Hopf代数联系。我们的表述虽然在很大程度上依赖于经典思想以及与Manzel和Schürmann上述工作密切相关的结果，但在若干点上仍具有原创性，并填补了非交换概率论文献中的空白。特别地，我们系统地使用代数群的语言和技术，结合shuffle群技术，证明了分别与自由、Boolean和单调概率理论自然关联的两种代数群概念是同一的。我们还获得了各种Hopf代数结构的显式公式，并详细阐述了文献中曾被隐去的具体论证。
英文摘要：
The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Schürmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and results in another construction of groups of characters encoding the behaviour of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Schürmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of noncrossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Schürmann's aforementioned work, is nevertheless original on several points and fills a gap in the non-commutative probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature.
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