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Crossed products and finitely presented groups. (English) Zbl 0978.16028
Let $$F$$ be a field, let $$n$$ be a positive integer and let $${\mathbf q}=(q_{ij})$$ be an $$n\times n$$ matrix with entries from $$F\setminus\{0\}$$ such that $$q_{ij}=q^{-1}_{ji}$$ and $$q_{ii}=1$$ for all $$i,j$$. Localised quantum $$n$$-space $$A({\mathbf q},F)$$ is the $$F$$-algebra generated by $$X^{\pm 1}_1,\ldots,X^{\pm 1}_n$$ subject to the relations $$X_iX_j=q_{ij}X_jX_i$$ for all $$i,j$$. These algebras are fundamental objects in the study of quantum groups; they also arise frequently in work on infinite groups and their group algebras, since $$A({\mathbf q},F)$$ may be regarded as the crossed product $$F*G$$ of a free Abelian group $$G$$ with $$F$$ central. Define $$t$$ to be the maximum of the ranks of the subgroups $$H$$ of $$G$$ for which $$F*H=FH$$, the ordinary group algebra. J. C. McConnell and J. J. Pettit [J. Lond. Math. Soc., II. Ser. 38, 47-55 (1988; Zbl 0652.16007)] observed that $$t$$ is a lower bound for the global and Krull dimensions of $$A({\mathbf q},F)$$, and conjectured that equality holds in both cases. The main theorem of this paper confirms both these equalities, a beautiful result which improves on earlier special cases obtained by a number of authors. The paper also includes applications to finitely presented groups which are somewhat too technical to state here.

##### MSC:
 16S35 Twisted and skew group rings, crossed products 20F05 Generators, relations, and presentations of groups 16E10 Homological dimension in associative algebras 16W35 Ring-theoretic aspects of quantum groups (MSC2000) 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 20F16 Solvable groups, supersolvable groups 16S34 Group rings 20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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