The Krull and global dimension of the tensor product of quantum tori (Q2816949)

A quantum torus is an associative algebra over a field \(k\) generated by elements \(x_1^{\pm 1},\ldots, x_n^{\pm 1}\) with defining relations \(x_ix_j=\lambda_{ij}x_jx_i\) for some nonzero parameters \(\lambda_{ij} \in k\). A quantum torus is a twisted group algebra \(k*A\), where \(A\) is a free abelian group of rank \(n\). \textit{J. C. McConnell} and \textit{J. J. Pettit} [J. Lond. Math. Soc., II. Ser. 38, No. 1, 47--55 (1988; Zbl 0652.16007)], showed that Krull and global dimensions \(d\) of these algebras coincide and \(1\leqslant d\leqslant n\). \textit{C. J. B. Brookes} [J. Group Theory 3, No. 4, 433--444 (2000; Zbl 0978.16028)], showed that \(d\) equals to supremum of ranks of subgroup \(B\) in \(A\) such that \(k*B\) is commutative. The present paper deals with the dimension of a tensor product of two algebras \(k*A_1 \) and \(k*A_2\). Suppose that \(\dim(k*A_i)<\mathrm{rk}(A_i)\) for both \(i\). Then the dimension of their tensor product does not exceed NEWLINE\[NEWLINE \min\left(\dim(k*A_1)+ \mathrm{rk}(A_2),\, \dim(k*A_2)+ \mathrm{rk}(A_2)\right)-1. NEWLINE\]NEWLINE In particular, if \(\dim(k*A_i)=\mathrm{rk}(A_i)-1\) for \(i=1,2\), then the dimension of the tensor product is equal to the sum of dimensions. Some results are obtained in the case when both algebras are central.