q-fundamental matrices are introduced and studied. Elementary operations on quantum matrices are discussed. The q-generalization of the classical Laplace's theorem is found, An application of the result is given.

In this paper, a su cient condition is given under which the smash product A#H is a trans nite left free normalizing extension of an algebra A. Moreover, the result is applied to a skew semigroup ring, a skew group ring and the quantum group Uq(sl(2)) such that some properties are shown.

In this paper, the concepts of a weak Hopf algebra and a quasi-braided almost bialgebra are introduced. It is shown that the quantum quasi-doubles of some weak Hopf algebras are quasi-braided almost bialgebras. This fact implies that some new solutions of the quantum Yang-Baxter equation can be constructed from some weak Hopf algebras, in particular, when the weak Hopf algebra is a finite Clifford monoid algebra.