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

TL;DR

This paper investigates how the Krull and global dimensions of n-dimensional quantum tori behave under tensor products, establishing bounds and conditions for additivity of these dimensions.

Contribution

It provides the first comprehensive analysis of the Krull and global dimensions of tensor products of quantum tori, including bounds and additivity conditions.

Findings

Derived a best possible upper bound for the dimension of tensor products of quantum tori.

Identified cases where the dimensions are additive with respect to tensoring.

Showed that the Krull and global dimensions coincide and relate to subgroup ranks.

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 $x_ix_j = q_{ij}x_jx_i$, 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 $\langle x_1, \cdots, x_n \rangle$ 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.