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Crossed products and multiplicative analogues of Weyl algebras. (English) Zbl 0652.16007
The subject of this article are certain algebras P(\(\lambda)\) that are generated over a field k by elements \(x_ 1,x_ 2,...,x_ n\) and their inverses \(x_ i^{-1}\) subject to the relations \(x_ ix_ jx_ i^{-1}x_ j^{-1}=\lambda_{ij}\), where \(\lambda_{ij}\in k\) \((1\leq i<j\leq n)\) are given. It is easy to see that the algebras \(P(\lambda)\) are Noetherian domains of Gelfand-Kirillov dimension n. They can be viewed as multiplicative analogs of the familiar Weyl algebras which are defined by assigning scalar values to the additive (Lie) commutators \(x_ ix_ j-x_ jx_ i\) \((1\leq i<j\leq n)\) of the generators \(x_ 1,x_ 2,...,x_ n\). The main results concern the Krull and global dimension of the algebras \(P(\lambda)\). For example, it is shown that if the subgroup of \(k^*\) generated by the scalars \(\lambda_{ij}\) has (maximum) rank \(1/2\;n(n-1)\), then \(P(\lambda)\) is a simple Noetherian domain of Krull and global dimension 1. Moreover, in this case, each simple \(P(\lambda)\)-module has Gelfand-Kirillov dimension n-1.
Reviewer: M.Lorenz

MSC:
16P40 Noetherian rings and modules (associative rings and algebras)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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