Binary trees, operads and Dykema’s T-transform in Free Probability

Abstract: In this talk, we shall discuss an operadic perspective on K. Dykema’s twisted factorization formula for the operator-valued T-transform in free probability. To begin with, we introduce in the general setting of an operad with multiplication two group products on formal series of operators, besides the one introduced by F. Chapoton. We explain how those products relate by means of certain transformation, that we call (abstract) T-transform, borrowing terminology from free probability. Specializing in the endomorphism operad gives a new perspective on the twisted factorization of the T-transform and to multiplicative free convolution. We will discuss connections to the work of A. Frabetti and C. Brouder by specializing our construction to the duoidal and dendriform operads.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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