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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 \(n=2\).

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
Full Text: DOI
[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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