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Finite-dimensional simple modules over quantised Weyl algebras. (English) Zbl 0923.16005
The quantised Weyl algebra $$A_n^{\overline q,\Lambda}$$ is an iterated skew polynomial ring in $$2n$$ indeterminates $$x_i$$ and $$y_i$$, $$1\leq i\leq n$$, involving parameters $$q_i$$, $$1\leq i\leq n$$, and $$\lambda_{ij}$$, $$1\leq i<j\leq n$$, which reduces to the usual Weyl algebra $$A_n$$ when all the parameters are set equal to $$1$$. It can be interpreted in terms of $$q_i$$-difference operators on a quantum space determined by the $$\lambda_{ij}$$’s and has been studied from the ring theoretical point of view by, among others, J. Alev and F. Dumas [J. Algebra 170, No. 1, 229-265 (1994; Zbl 0820.17015)], and the reviewer [J. Algebra 174, No. 1, 267-281 (1995; Zbl 0833.16025)]. Here, using a result on the case $$n=1$$ due to the reviewer [J. Pure Appl. Algebra 98, No. 1, 45-55 (1995; Zbl 0829.16017)], the author classifies the finite-dimensional simple $$A_n^{\overline q,\Lambda}$$-modules when $$q_1$$ is not a root of unity and either each $$\lambda_{ij}=1$$ or, for each $$j\geq 2$$, neither $$\lambda_{1j}$$ nor $$q_1\lambda_{1j}$$ is a root of unity or $$n=2$$.

##### MSC:
 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras 16S36 Ordinary and skew polynomial rings and semigroup rings 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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##### References:
 [1] Jordan, J. Pure Appl. Algebra 98 pp 45– (1995) · Zbl 0829.16017 [2] DOI: 10.1016/S0021-8693(05)80036-5 · Zbl 0779.16010 [3] DOI: 10.1006/jabr.1995.1128 · Zbl 0833.16025 [4] DOI: 10.1006/jabr.1994.1336 · Zbl 0820.17015 [5] DOI: 10.1007/BF01218386 · Zbl 0651.17008 [6] DOI: 10.1006/jabr.1995.1276 · Zbl 0846.17007
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