On bialgebras associated with paths and essential paths on ADE graphs
摘要
We define a graded multiplication on the vector space of essential paths on a graph (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.
引用
@article{arxiv.math-ph/0411048,
title = {On bialgebras associated with paths and essential paths on ADE graphs},
author = {Robert Coquereaux and Ariel O. Garcia},
journal= {arXiv preprint arXiv:math-ph/0411048},
year = {2015}
}