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.