Spectrum of the quantized Weyl algebra. (Spectre de l’algèbre de Weyl quantique.) (French. English summary) Zbl 0876.17012

Let \(k\) be a field of characteristic 0, let \(n\) be a positive integer, let \({\mathbf q} =(q_1, \dots, q_n) \in(k^*)^n\) and let \(\Lambda= (\lambda_{ij})\) be an \(n\times n\) matrix whose entries are non-zero elements of \(k\) satisfying \(\lambda_{ij} =\lambda_{ij}^{-1}\) and \(\lambda_{ii} =1\) for all \(i\) and \(j\). Then \(n\)th quantized Weyl algebra \(A_n^{\underline{q}^Lambda}\) is the \(k\)-algebra on the \(2n\) generators \(x_1, \dots, x_n,\) \(y_1, \dots, y_n\), subject to the relations (1) for all \(i\) and \(j\) with \(i<j\), \(x_ix_j= \lambda_{ij} q_ix_jx_i\), \(y_iy_j= \lambda_{ij} y_jy_i\), \(x_iy_j= \lambda^{-1}_{ij} y_jx_i\), \(y_ix_j= \lambda^{-1}_{ij} q_i^{-1} x_jy_i\), and (2) for all \(i, x_iy_i- q_iy_ix_i=1 +\sum^{i-1}_{j=1} (q_j-1) y_jx_j\). This algebra was defined by G. Maltsiniotis as the algebra of differential operators on quantum \(n\)-space, and studied in, for example J. Alev and F. Dumas [J. Algebra 170, 229-265 (1994; Zbl 0820.17015)].
The main purpose of this paper is to describe the prime spectrum of \(A_n^{\underline{q}^Lambda}\) in the case where the subgroup of \(k^*\) generated by \(\{\lambda_{ij}, q_\ell: 1<i, j,\ell \leq n\}\) is torsion free of rank \({1\over 2} n(n+1)\). It is proved that under these hypotheses the spectrum is the union of a 1-torus of maximal ideals (the kernels of the algebra homomorphisms to \(k)\) together with a finite and explicitly described set of prime ideals. As a corollary it is deduced that, under the same hypotheses, the group of \(k\)-algebra automorphisms of \(A_n^{\underline{q}^Lambda}\) is an \(n\)-torus, (where \(\alpha= (\alpha_1, \dots, \alpha_n) \in(k^*)^n\) sends \(x_i\) to \(\alpha_ix_i\) and \(y_i\) to \(\alpha_i^{-1}y_i)\). A similar description of the prime spectrum of \(A_n^{\underline{q}^\Lambda}\), (under somewhat weaker restrictions on the parameters) has been independently obtained in [M. Akhavizadegan and D. A. Jordan, Glasg. Math. J. 38, 283-297 (1996)].


17B37 Quantum groups (quantized enveloping algebras) and related deformations
16S32 Rings of differential operators (associative algebraic aspects)
16D25 Ideals in associative algebras


Zbl 0820.17015
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