Hopf algebras and the logarithm of the 𝑆-transform in free probability

Source abstract

Let k k be a positive integer and let G k \mathcal {G}_k denote the set of all joint distributions of k k -tuples ( a 1 , … , a k ) (a_1,\ldots ,a_k) in a noncommutative probability space ( A , φ ) (\mathcal {A},\varphi ) such that φ ( a 1 ) = ⋯ = φ ( a k ) = 1 \varphi (a_1)=\cdots =\varphi (a_k) = 1 . G k \mathcal {G}_k is a group under the operation of the free multiplicative convolution ⊠ \boxtimes . We identify ( G k , ⊠ ) \bigl (\,\mathcal {G}_k, \boxtimes \, \bigr ) as the group of characters of a certain Hopf algebra Y ( k ) \mathcal {Y}^{(k)} . Then, by using the log map from characters to infinitesimal characters of Y ( k ) \mathcal {Y}^{(k)} , we introduce a transform L S μ LS_{\mu } for distributions μ ∈ G k \mu \in \mathcal {G}_k . L S μ LS_{\mu } is a power series in k k noncommuting indeterminates z 1 , … , z k z_1, \ldots , z_k ; its coefficients can be computed from the coefficients of the R R -transform of μ \mu by using summations over chains in the lattices N C ( n ) NC(n) of noncrossing partitions. The L S LS -transform has the “linearizing” property that $$L S μ ⊠ ν = L S μ + L S ν , ∀ μ , ν ∈ G k such that μ ⊠ ν = ν ⊠ μ . LS_{\mu \boxtimes \nu } =LS_{\mu } +LS_{\nu }, \ \ \forall \, \mu , \nu \in \mathcal {G}_k \mbox { such that } \mu \boxtimes \nu = \nu \boxtimes \mu .$$ In the particular case k = 1 k=1 one has that Y ( 1 ) {\mathcal Y}^{(1)} is naturally isomorphic to the Hopf algebra Sym \mbox {Sym} of symmetric functions and that the L S LS -transform is very closely related to the logarithm of the S S -transform of Voiculescu by the formula $$L S μ ( z ) = − z log ⁡ S μ ( z ) , ∀ μ ∈ G 1 . LS_{\mu } (z) = -z \log S_{\mu } (z), \ \ \forall \, \mu \in \mathcal {G}_1.$$ In this case the group ( G 1 , ⊠ ) (\mathcal G_1, \boxtimes ) can be identified as the group of characters of Sym \mbox {Sym} , in such a way that the S S -transform, its reciprocal 1 / S 1/S and its logarithm log ⁡ S \log S relate in a natural sense to the sequences of complete, elementary and, respectively, power sum symmetric functions.