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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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