zbMATH — the first resource for mathematics

Spectrum of the quantized Weyl algebra. (Spectre de l’algĂ¨bre 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)].

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