  We study finite quasi-quantum groups in their quiver setting developed
recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain
a classification of finite-dimensional pointed Majid algebras of finite
corepresentation type, or equivalently a classification of elementary
quasi-Hopf algebras of finite representation type, over the field of complex
numbers. By the Tannaka-Krein duality principle, this provides a classification
of the finite tensor categories in which every simple object has
Frobenius-Perron dimension 1 and there are finitely many indecomposable objects
up to isomorphism. Some interesting information of these finite tensor
categories is given by making use of the quiver representation theory.
