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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.

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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