Self-Dual Hopf Quivers

ABSTRACT

We study self-dual coradically graded pointed Hopf algebras with a help of the dual Gabriel theorem for pointed Hopf algebras (van Oystaeyen and Zhang, 2004). The co-Gabriel Quivers of such Hopf algebras are said to be self-dual. An explicit classification of self-dual Hopf quivers is obtained. We also prove that finite dimensional pointed Hopf algebras with self-dual graded versions are generated by group-like and skew-primitive elements as associative algebras. This partially justifies a conjecture of Andruskiewitsch and Schneider (2000) and may help to classify finite dimensional self-dual coradically graded pointed Hopf algebras.

Keywords:

• Hopf algebra
• Quiver

Mathematics Subject Classification:

• Primary 16W30
• Secondary 16W70, 16G20

ACKNOWLEDGMENT

The authors are very grateful to the referee for his/her valuable suggestions and comments.

Communicated by C. Cibils.