Gröbner–Shirshov pair of irreducible modules over quantized enveloping algebra of type An

In this paper, first, we give a Gröbner–Shirshov pair of finite-dimensional irreducible module [Formula: see text] over [Formula: see text] the quantized enveloping algebra of type [Formula: see text] by using the double free module method and the known Gröbner–Shirshov basis of [Formula: see text] Then, by specializing a suitable version of [Formula: see text] at [Formula: see text] we get a Gröbner–Shirshov basis of [Formula: see text] and get a Gröbner–Shirshov pair for the finite-dimensional irreducible module [Formula: see text] over [Formula: see text].