Abstract: By extending the theory of Killing form and adjoint representation over Hopf algebras to the weak Hopf algebras, the author gave the concepts and properties of Killing form and adjoint representation over weak Hopf algebras, and discussed the adjoint representations and Killing form of a weak Hopf algebras H₈,  so as to realize the application of the theory of Killing form and adjoint representation over the weak Hopf algebra.

Key words: weak Hopf algebra, Killing form, adjoint representation, Killing radical