A quantum analog of the Poincare-Birkhoff-Witt theorem

We reduce the basis construction problem for Hopf algebras generated by skew-primitive semi-invariants to a study of special elements, called ``super-letters,'' which are defined by Shirshov standard words. In this way we show that above Hopf algebras always have sets of PBW-generators (``hard'' super-letters). It is shown also that these Hopf algebras having not more than finitely many ``hard'' super-letters share some of the properties of universal enveloping algebras of finite-dimensional Lie algebras. The background for the proofs is the construction of a filtration such that the associated graded algebra is obtained by iterating the skew polynomials construction, possibly followed with factorization.