De Finetti Theorems for the Unitary Dual Group

Symmetry Integrability and Geometry-Methods and Applications

article

Laura Maassen¹  Moritz Weber²  Isabelle Baraquin³  Guillaume Cébron⁴  Uwe Franz³ 

[1] Formerly: RWTH Aachen University;Saarland University;Laboratoire de mathématiques de Besançon, UMR 6623, CNRS, Université Bourgogne Franche-Comté;Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse

关键词: de Finetti theorem;    distributional invariance;    exchangeable;    Brown algebra;    unitary dual group;    $R$-diagonal elements;    free circular elements.;   

DOI  :  10.3842/SIGMA.2022.067

来源: National Academy of Science of Ukraine

【 摘 要 】

We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing $R$-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in $W^*$-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group $U_n^+$. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in $W^*$-probability spaces. On the other hand, if we drop the assumption of faithful states in $W^*$-probability spaces, we obtain a non-trivial half a de Finetti theorem similar to the case of the dual group action.

【 授权许可】

Unknown