The stable category of graded MCM modules over S ⋉ S_σ(-1) is triangle equivalent to the bounded derived category of modules over the Koszul dual of the Zhang twist S^{σ^{-1}}.
Gaddis,The isomorphism problem for quantum affine spaces, homogenized quantized Weyl algebras, and quan- tum matrix algebras, J
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Trivial extensions of Koszul Artin-Schelter regular algebras
The stable category of graded MCM modules over S ⋉ S_σ(-1) is triangle equivalent to the bounded derived category of modules over the Koszul dual of the Zhang twist S^{σ^{-1}}.