Classification and dilation for $q$-commuting $2\times 2$ scalar matrices

A tuple $\underset{‾}{T}=(T_1,\dots ,T_k)$ of operators on a Hilbert space $\mathcal{H}$ is said to be $q\mathit{\text{-commuting}}\mathit{\text{with}}$ $\mathrm{\parallel }q\mathrm{\parallel }=1$ or simply $q$-$\mathit{\text{commuting}}$ if there is a family of scalars $q=\{q_{ij}\in \mathbb{C}:\mathrm{\mid }{q}_{ij}\mathrm{\mid }=1,q_{ij}=q_{ji}^{-1},1\le i<j\le k\}$ such that $T_i{T}_j=q_{ij}{T}_j{T}_i$ for $1\le i<j\le k$. Moreover, if each $q_{ij}=-1$, then $\underset{‾}{T}$ is called an $\mathit{\text{anti-commuting}}\mathit{\text{tuple}}$. A well-known result due to Holbrook \cite{Holbrook} states that a commuting $k$-tuple consisting of $2\times 2$ scalar matrix contractions always dilates to a commuting $k$-tuple of unitaries for any $k\ge 1$. To find a generalization of this result for a $q$-commuting $k$-tuple of $2\times 2$ scalar matrix contractions, we first classify such tuples into three types upto similarity. Then we prove that a $q$-commuting tuple which is unitarily equivalent to any of these three types, admits a $\tilde{q}$-unitary dilation, where $\tilde{q}\subseteq q\cup \{1\}$. A special emphasis is given to the dilation of an anti-commuting tuple of $2\times 2$ scalar matrix contractions.