The second quantum Weyl algebra at roots of unity has PI degree n1 n2 times ord(λ^{n1 n2}), and its simple modules split into three torsion types, each explicitly constructed and classified.
In [18, Theorem 3.1], we determined the PI degree under the divisibil- ity condition
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Simple Modules over Second Quantum Weyl Algebra
The second quantum Weyl algebra at roots of unity has PI degree n1 n2 times ord(λ^{n1 n2}), and its simple modules split into three torsion types, each explicitly constructed and classified.