On bialgebras associated with paths and essential paths on ADE graphs

TL;DR

This paper introduces a new associative graded multiplication on essential paths of ADE graphs, leading to novel bialgebra structures related to quantum groupoids, expanding the algebraic understanding of these graph-based systems.

Contribution

It defines a new graded associative product on essential paths and constructs a weak bialgebra directly, without relying on the existing Double Triangle Algebra framework.

Findings

Established an associative graded multiplication on essential paths.

Constructed a new weak bialgebra structure from the paths.

Connected the new structure to quantum groupoid concepts.

Abstract

We define a graded multiplication on the vector space of essential paths on a graph $G$ (a tree) and show that it is associative. In most interesting applications, this tree is an ADE Dynkin diagram. The vector space of length preserving endomorphisms of essential paths has a grading obtained from the length of paths and possesses several interesting bialgebra structures. One of these, the Double Triangle Algebra (DTA) of A. Ocneanu, is a particular kind of quantum groupoid (a weak Hopf algebra) and was studied elsewhere; its coproduct gives a filtrated convolution product on the dual vector space. Another bialgebra structure is obtained by replacing this filtered convolution product by a graded associative product.It can be obtained from the former by projection on a subspace of maximal grade, but it is interesting to define it directly, without using the DTA. What is obtained is a weak bialgebra, not a weak Hopf algebra.