On bialgebras associated with paths and essential paths on ADE graphs

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.

References in corpus (2)

• The many faces of Ocneanu cells
• On quantum symmetries of ADE graphs

Cited by in corpus (4)

• Racah - Wigner quantum 6j Symbols, Ocneanu Cells for AN diagrams, and quantum groupoids
• Orders and dimensions for sl(2) or sl(3) module categories and Boundary Conformal Field Theories on a torus
• McKay correspondence and equivariant sheaves on $P^1$
• Essential paths space on ADE SU(3) graphs: A geometric approach