scholarly journals The unitary implementation of a measured quantum groupoid action

2010 ◽  
Vol 17 (2) ◽  
pp. 233-302
Author(s):  
Michel Enock
Keyword(s):  
Quantum Groupoid

Quantum Groupoid

2004 ◽  
pp. 322-322
Author(s):  
Jian-zu Zhang ◽  
Roberto Floreanini ◽  
Steven Duplij ◽  
Steven Duplij ◽  
Dmitri Gitman ◽  
...  
Keyword(s):  
Quantum Groupoid

scholarly journals ON BIALGEBRAS ASSOCIATED WITH PATHS AND ESSENTIAL PATHS ON ADE GRAPHS

2005 ◽  
Vol 02 (03) ◽  
pp. 441-466 ◽  
Author(s):  
R. COQUEREAUX ◽  
A. O. GARCÍA
Keyword(s):  
Hopf Algebra ◽  
Vector Space ◽  
Dynkin Diagram ◽  
Weak Hopf Algebra ◽  
Convolution Product ◽  
Dual Vector ◽  
Associative Product ◽  
Quantum Groupoid

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.

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