Article Dans Une Revue (Article De Synthèse) Infinite Dimensional Analysis, Quantum Probability and Related Topics Année : 2024

Algebraic groups in non-commutative probability theory revisited

Résumé

The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Sch\"urmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and resulting 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 Sch\"urmann, 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 Sch\"urmann's aforementioned work, is nevertheless original on several points and fills a gap in the free 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.

Dates et versions

hal-03806588 , version 1 (07-10-2022)

Identifiants

Citer

Ilya Chevyrev, Kurusch Ebrahimi-Fard, Frédéric Patras. Algebraic groups in non-commutative probability theory revisited. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2024, 27 (03), ⟨10.1142/S021902572450005X⟩. ⟨hal-03806588⟩

Collections

78 Consultations
0 Téléchargements

Altmetric

Partager

  • More