The krull and global dimension of the tensor product of n-dimensional quantum tori

Abstract

The n-dimensional quantum torus is defined as the $F$-algebra generated by variables $x_1,\cdots ,x_n$ together with their inverses satisfying the relations $x_i{x}_j=q_{ij}{x}_j{x}_i$, where $q_{ij}\in F$. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of $⟨x_1,\cdots ,x_n⟩$ that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori %over the base field. We derive a best possible upper bound for the dimension of such a tensor product and %deduce from this special cases in which the dimension is additive with respect to tensoring.