Representations of Weak Hopf Algebras Associated to Cyclic Quivers

ABSTRACT

The aim of this paper is to study representations of weak Hopf algebras. We first prove that the global dimension of a weak Hopf algebra is the same as the projective dimension of its left unital subalgebra and deduce some equivalent conditions for its semisimplicity. Then we give the construction of a typical example 𝔲(m, L) of finite dimensional semisimple weak Hopf algebra, which associates to a cyclic quiver . All irreducible modules of 𝔲(m, L) are classified. The structure of the truncated tensor product for any two irreducible 𝔲(m, L)-modules V and W is obtained.

Key Words:

• Cyclic quiver
• Fusion rule
• Trivial module
• Weak Hopf algebra

Mathematics Subject Classification:

• Primary 16D70
• Secondary 16W30

ACKNOWLEDGMENTS

Dingguo Wang was supported by the Young Academic Backbone Foundation of Shandong Province and also the Project-sponsored by SRF for ROCS, SEM. Shilin Yang was partially supported by the National Science Foundation of China (10271014) and Natural Science Foundation of Beijing City (1042001).

The authors thank the referee very much for bringing a reference Andruskiewitsch and Natale (2005) to their attention, for pointing out some errors in the first and second version of the present article, and for useful comments.

Notes

Communicated by R. Wisbauer.