Simple modules over 3-cyclic quantum Weyl Algebra at roots of unity

This article undertakes an exploration of simple modules of 3-cyclic quantum Weyl algebra at roots of unity. Under the roots of unity assumption, the algebra becomes a Polynomial Identity algebra and the vector space dimension of the simple modules is bounded above by its PI degree. The article systematically classifies all potential simple modules and computes the algebra's center.
Comment: 23 pages. Comments are welcomed