ProPs of graphs and generalised traces

Pierre J. Clavier (1) , Loïc Foissy (2) , Sylvie Paycha (3)

1 SEPAREX
2 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

We assign generalised convolutions (resp. traces) to graphs whose edges are decorated by smooth kernels (resp. smoothing operators) on a closed manifold. To do so, we introduce the concept of TraPs (Traces and Permutations), which roughly correspond to ProPs (Products and Permutations) without vertical concatenation and equipped with families of generalised partial traces. They can be equipped with a ProP structure in deriving vertical concatenation from the partial traces and we relate TraPs to wheeled ProPs first introduced by Merkulov. We further build their free object and give precise proofs of universal properties of ProPs and TraPs.

Mots clés: ProP; trace; distribution kernel; convolution

Domaines: Combinatoire [math.CO]; Algèbres d'opérateurs [math.OA]; Physique mathématique [math-ph]

Pierre J. Clavier, Loïc Foissy, Sylvie Paycha. ProPs of graphs and generalised traces. 2020. ⟨hal-02563704⟩