On bialgebras associated with paths and essential paths on ADE graphs

Robert Coquereaux (1, 2) , Ariel O. Garcia (3, 4)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332
3 FORSCHUNGSZENTRUM KARLSRUHE - Forschungszentrum Karlsruhe
4 MPI-P - Max-Planck-Institut für Physik (Werner-Heisenberg-Institut)

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.

Mots clés: quantum symmetries; bialgebras; ADE; graphs; conformal field theories

Domaines: Physique mathématique [math-ph]; Physique des Hautes Energies - Théorie [hep-th]; Algèbres quantiques [math.QA]

Robert Coquereaux, Ariel O. Garcia. On bialgebras associated with paths and essential paths on ADE graphs. International Journal of Geometric Methods in Modern Physics, 2005, 3 (2), pp.1. ⟨hal-00003260⟩