quantum groupoid
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2005 ◽  
Vol 02 (03) ◽  
pp. 441-466 ◽  
Author(s):  
R. COQUEREAUX ◽  
A. O. GARCÍA

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.


Author(s):  
Jian-zu Zhang ◽  
Roberto Floreanini ◽  
Steven Duplij ◽  
Steven Duplij ◽  
Dmitri Gitman ◽  
...  
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