Rank 6 Quantum Torus and Dimension Theory

• The Rank 6 Quantum Torus is a noncommutative Laurent algebra defined by six invertible generators and multiparameter relations, establishing its structure.
• Its dimension equals the maximal rank of a free abelian subgroup with trivial pairwise commutation, offering insights into its representation and module theories.
• Analyzed via a clique-based method on its commutation graph, its tensor product behavior reveals the interplay between commutativity and noncommutativity.

A rank 6 quantum torus is the multiparameter noncommutative Laurent algebra $A_6(Q)$ over a base field $F$, with six invertible generators $x_1^{\pm1},\ldots,x_6^{\pm1}$ and relations %%%%3%%%%, where $Q = (q_{ij}) \in (F^\times)^{6\times 6}$ is a multiplicatively antisymmetric parameter matrix satisfying $q_{ii}=1$ and $q_{ij}q_{ji}=1$ for all $i,j$. The global (left/right) and Krull dimensions of $A_6(Q)$ both equal the maximal rank of a free abelian subgroup $H \leq \mathbb{Z}^6$ for which all pairwise commutators $q_{ij}$ restrict trivially, i.e., $F*H$ is commutative. This rank serves as a fundamental invariant controlling several aspects of the representation theory and the structure of simple modules of $A_6(Q)$ (Gupta, 2014).

1. Formal Definition and Algebraic Structure

The n-dimensional quantum torus $A_n(Q)$ is defined by selecting a field $F$ and a free abelian group $A \cong \mathbb{Z}^n$ with ordered basis $e_1,\ldots,e_n$. To each pair $(i,j)$, assign a parameter $q_{ij} \in F^\times$ with $q_{ii}=1$ and $q_{ij}q_{ji}=1$. The algebra is

$$A_n(Q) = F\langle x_1^{\pm1}, \ldots, x_n^{\pm1} \mid x_ix_j = q_{ij}x_jx_i,\, x_i^{-1}x_i=1\rangle.$$

Alternatively, $A_n(Q) \cong F* A$ is the twisted group algebra with basis $\{\bar{a}: a \in A\}$ and multiplication $\bar{e}_i\bar{e}_j = q_{ij}\overline{e_i + e_j}$. Subgroups $H\leq A$ inherit this structure, and the subalgebra on $\{x^h: h\in H\}$ is itself a quantum torus $F*H$. $H$ is commutative for $Q$ precisely when $q_{ij}=1$ for $i, j$ in any basis of $H$.

2. Dimension Theorem: Coincidence of Krull and Global Dimension

The single most important result concerning $A_6(Q)$—and all quantum tori—is that the Krull and global ring-theoretic dimensions coincide, given by: $$\dim A_6(Q) := \operatorname{Kdim}A_6(Q) = \operatorname{gldim}A_6(Q) = \sup\{ \operatorname{rank}(H) : H\leq \mathbb{Z}^6,\, F*H\,\textrm{commutative} \}.$$ The maximal dimension is 6, achieved precisely when $Q$ is the identity and $A_6(Q)\cong F[x_1^{\pm1},\ldots,x_6^{\pm1}]$. Lower values correspond to increasingly noncommutative "twist" matrices $Q$. The proof establishes lower bounds by localization to commutative subtori and upper bounds by finding simple modules with minimal Gelfand–Kirillov dimension and establishing the link to commutative subgroups (Gupta, 2014).

3. Explicit Computation for n = 6

For $n=6$, the dimension of $A_6(Q)$ is determined by the size of the largest commutative subgroup $H\leq \mathbb{Z}^6$, equivalently, the largest clique in the commutation graph (vertices $1,\ldots,6$, edge $i$–$j$ if $q_{ij}=1$).

Representative Examples:

Example Description	Rank of $H$	Dimension $\dim A_6(Q)$	Conditions on $Q$
Generic parameters	1	1	All $q_{ij}$ multiplicatively independent, $q_{ij}\neq 1$
One commuting pair	2	2	$q_{12}=1$, all others generic ($\neq1$)
Single 3-clique	3	3	$q_{ij}=1$, $1\leq i<j\leq3$, others generic
Two disjoint 3-cliques	3	3	$q_{ij}=1$ for $1\leq i<j\leq3$, or $4\leq i<j\leq6$
Fully commutative	6	6	All $q_{ij}=1$

For generic $Q$, only cyclic subgroups can be commutative, so $\dim A_6(Q)=1$. For certain specializations (e.g., some $q_{ij}=1$), larger commutative subtori exist, increasing the dimension up to the maximal value 6. In all cases, $\dim A_6(Q)$ is the size of the largest set of pairwise-commuting generators.

4. Analysis of Commutative Subgroups and Clique Structure

The determination of $\dim A_6(Q)$ reduces, for $Q$ given, to a combinatorial analysis of the commutation graph on $\{1,\ldots,6\}$: vertices $i$ and $j$ are connected if $q_{ij}=1$. The maximal size of a clique in this graph is the rank of the largest commutative subgroup $H$ and gives the global/Krull dimension.

This correspondence provides a direct computational approach: construct the commutation graph, enumerate maximal cliques, and select the one of greatest cardinality.

It follows that, except in degenerate parameter cases, rank 6 quantum tori can have dimensions anywhere from 1 up to 6, depending on the multiplicative relations among the $q_{ij}$.

5. Behavior of Dimension Under Tensor Products

Given two tori $A_n(Q)$ and $A_m(Q')$, their tensor product is itself a quantum torus: $$A_n(Q)\otimes_F A_m(Q') \cong A_{n+m}(\operatorname{diag}(Q,Q')).$$ Dimension theory for tensor products is governed by several results [(Gupta, 2014), Theorem 5.7]:

• Super-additivity: $\dim (A_n\otimes A_m) \geq \dim A_n+\dim A_m$.
• Upper bound: Provided neither factor is maximally commutative,

$$\dim\left(A_n\otimes A_m\right) \leq \min\{ \dim A_n + m,\, \dim A_m + n \} - 1.$$

• Additivity criteria: If at least one factor is virtually commutative (i.e. $\dim = n$ or $m$), or if codimension $\leq1$, then

$\dim(A_n\otimes A_m)=\dim A_n+\dim A_m.$

• Corollaries 5.11 and 5.15 in (Gupta, 2014) provide necessary and sufficient conditions for equality in the inequality above.

For explicit illustration, $$A_3(q^{(1)})\otimes A_3(q^{(2)}) \cong A_6(\operatorname{diag}(q^{(1)},q^{(2)}))$$ satisfies

$$\dim A_3 + \dim A_3 \leq \dim A_6 \leq 2\cdot 3 - 1 = 5,$$

with equality if, for example, one factor has codimension 1.

6. Related Results, References, and Further Context

The formulation and dimension theory of quantum tori outlined here are situated within the broader theory of twisted group algebras and noncommutative Laurent-type algebras. Key references include McConnell–Pettit (1988) for crossed product analogues, Brookes (2000) for twisted group algebra context, Brookes–Groves (2000, 2002) for module theory over crossed products, and Wadsley (2005) for geometric invariants. The main dimension results, tensor product bounds, and combinatorial reductions to clique size are due to Brookes, Groves, and Wadsley (Gupta, 2014).

A plausible implication is that any rank 6 quantum torus can arise as a tensor product of lower-rank tori using appropriate parameter matrices, but the resulting dimension depends sensitively on the interaction between the commutative subgroups of the factors and the global structure of the commutation graph. This clique-theoretic perspective provides a practical route for computing key invariants and for constructing examples in the theory of noncommutative algebras.