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Quantum Weyl algebras. (English) Zbl 0846.17007
Let $$K$$ be a field, $$V$$ an $$n$$-dimensional vector space over $$K$$, and $$R: V\otimes V\to V\otimes V$$ a Hecke symmetry with respect to some $$q\in K^*$$. J. Wess and B. Zumino have constructed a quantization $$A_n (R)$$ of the $$n$$-th Weyl algebra $$A_n$$ based on $$R$$, which may be viewed as the algebra of quantized differential operators on the $$R$$-symmetric algebra [Nucl. Phys. B, Proc. Suppl. 18, 302-312 (1990)]. Here the authors study some ring-theoretic properties of $$A_n (R)$$, showing in particular that it is left and right primitive whenever $$q$$ is not a root of unity and is nonsimple whenever it is infinite-dimensional and $$q\neq \pm1$$. They also show that under some assumptions on $$R$$ this algebra $$A_n (R)$$ is an Auslander regular Cohen-Macaulay Noetherian domain with GK dimension $$2n$$. It may be regarded as a formal deformation of $$A_n$$ in the sense of Gerstenhaber.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16S32 Rings of differential operators (associative algebraic aspects)
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