Let A be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra g. We present a construction of the Mickelsson algebra Z(A, g) relative to the left ideal in A generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum g-modules. We give an explicit expression for a PBW basis in Z(A, g) in the case when A = U (a) of a finite-dimensional Lie algebra a ⊃ g. For A = U q (a) and g the commutant of a Levi subalgebra in a, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to C[[ℏ]].
AMS classification codes: 17B10, 17B37.-