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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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