Towards a Classification of Multi-Faced Independence: A Representation-Theoretic Approach

Abstract

We attack the classification problem of multi-faced independences, the first non-trivial example being Voiculescu's bi-freeness. While the present paper does not achieve a complete classification, it formalizes the idea of lifting an operator on a pre-Hilbert space in a "universal" way to a larger product space, which is key for the construction of (old and new) examples. It will be shown how universal lifts can be used to construct very well-behaved (multi-faced) independences in general. Furthermore, we entirely classify universal lifts to the tensor product and to the free product of pre-Hilbert spaces. Our work brings to light surprising new examples of 2-faced independences. Most noteworthy, for many known 2-faced independences, we find that they admit continuous deformations within the class of 2-faced independences, showing in particular that, in contrast with the single faced case, this class is infinite (and even uncountable).

Categorial Independence and Lévy Processes

“…For all of these categories one can define (multivariate) universal products, and develop the theory of Lévy processes, as discussed in this paper, as well as other topics related to independence, as has been successfully done with cumulants in [32]. In the last years, such multivariate independences have received a lot of attention and many more examples have been found and studied to some extent [13,14,16,21,22,23,29,30]. The richness of examples when compared to the univariate case underlines the value of general methods applying to all multivariate universal independences at once.…”

“…[6,7]). Also, following Voiculescu's invention of bifreeness [44], many examples of multivariate independences have been exhibited [13,16,21,22,23,29,30] and some general theory has been developed [14,32], and all those independences have associated classes of Lévy processes that are very interesting to study. The main aim of these notes is to give a unified approach to these different situations.…”

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

“…With the works of Skoufranis, Gu, Hasebe, Gerhold, Liu, [14,19,20,21,27], several more instances of two-faced independences besides bifreeness emerged, mixing one type of independence (free, monotone, anti-monotone, boolean, or tensor) for left-sided random variables with another one for right-sided random variables (Gerhold, Hasebe and Ulrich in [16] and Gerhold and Varšo in [17], see also [39], even establish continuous families of two-faced independences, including nontrivial bi-tensor independences). For most of these independences, cumulants have been defined by exhibiting a certain poset of bipartitions and applying the usual machinery of Rota's combinatorics and Möbius inversion.…”

“…The combinatorial structures we study here only lead to trivial bi-tensor independence, which coincides with usual tensor independence because the full set of partitions of a translucent set is not constrained by colouration. Non-trivial bi-tensor independences, such as appear in[16] and[39] as deformations are beyond our scope here.…”

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

“…The axiomatic framework has been adapted to the two-faced (and more generally multi-faced-multi-state) case by Manzel and Schürmann [MS17]. With the works of Skoufranis, Gu, Hasebe, Gerhold, Liu, [GS17, GS19, GHS20, Ger17, Liu19], several more instances of two-faced independences besides bifreeness emerged, mixing one type of independence (free, monotone, anti-monotone, boolean or tensor) for left-sided random variables and another one for right-sided random variables (in [GHU21], Gerhold, Hasebe and Ulrich even establish continuous families of two-faced independences). For most of these independences, cumulants have been defined by exhibiting a certain poset of bipartitions and applying the usual machinery.…”