Hopf algebras and the logarithm of the S-transform in free probability – Extended abstract
نویسندگان
چکیده
Two basic tools of free probability are the R-transform and the S-transform. These transforms were introduced by Voiculescu in the 1980s, and are used to understand the addition and the multiplication of two free random variables respectively. The R-transform has a natural and very useful multi-variable extension describing the addition of two free k-tuples of random variables, but the problem of finding such an extension for the S-transform is open. The problem of the multi-variable S-transform can be re-phrased as the problem of understanding the structure of the group (Gk, ), where Gk is a special set of joint distributions of noncommutative k-tuples (see precise definition in Equation (3.3) below), and where (“free multiplicative convolution”) is a binary operation on Gk which encodes the multiplication of free k-tuples. At present, the structure of (Gk, ) is well-understood only in the special case k = 1; in this case, the S-transform of Voiculescu provides an isomorphism between G1 and a multiplicative group of power series in one variable. (A word of caution here: G1 is commutative, but it is easy to see that Gk is not commutative for any k ≥ 2.) In [4] we use Hopf algebra methods in order to study the multiplication of free k-tuples. Specifically, we construct a combinatorial Hopf algebra Y such that (Gk, ) is naturally isomorphic to the the group X(Y) of characters of Y. We then employ the log map from characters to infinitesimal characters of Y, to introduce a transform LSμ for distributions μ ∈ Gk. LSμ is a power series in k non-commuting indeterminates z1, . . . , zk; its coefficients can be computed from the coefficients of the R-transform of μ †Research supported by a Discovery Grant from NSERC, Canada.
منابع مشابه
Hopf algebras and the logarithm of the S-transform in free probability
Let k be a positive integer and let Gk denote the set of all joint distributions of k-tuples (a1, . . . , ak) in a non-commutative probability space (A, φ) such that φ(a1) = · · · = φ(ak) = 1. Gk is a group under the operation of free multiplicative convolution ⊠. We identify ( Gk,⊠ ) as the group of characters of a certain Hopf algebra Y. Then, by using the log map from characters to infinites...
متن کاملAdjunctions between Hom and Tensor as endofunctors of (bi-) module category of comodule algebras over a quasi-Hopf algebra.
For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor endofunctors V k - and - kV are left adjoint to some kinds of Hom-endofunctors of _HM. The units and counits of these adjunctions are formally trivial as in the classical case.The category of (bi-) modules over a quasi-Hopf algebra is monoidal and some generalized versions of Hom-tensor relations have been st...
متن کاملGorenstein global dimensions for Hopf algebra actions
Let $H$ be a Hopf algebra and $A$ an $H$-bimodule algebra. In this paper, we investigate Gorenstein global dimensions for Hopf algebras and twisted smash product algebras $Astar H$. Results from the literature are generalized.
متن کاملOn the cyclic Homology of multiplier Hopf algebras
In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ consisting of a regular multiplier Hopf algebra $mathcal{H}$, a left $mathcal{H}$-comodule algebra $mathcal{R}$, and a unital left $mathcal{H}$-module $mathcal{X}$ which is also a unital algebra. First, we construct a para...
متن کامل