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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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