Intertwining of $\ast$-regular $q$-isometric dilations

A tuple $\underset{‾}{T}=(T_1,\dots ,T_k)$ of contractions on a Hilbert space $\mathcal{H}$ is said to be $q$-commuting with $\mathrm{\parallel }q\mathrm{\parallel }=1$ if there exists 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 $1\le i<j\le k$. In this article, we characterize $q$-commuting pairs of contractions with $\mathrm{\parallel }q\mathrm{\parallel }=1$ that admit a minimal $\ast$-regular $q$-isometric dilation. We present sufficient conditions for the commutant lifting theorem for such pairs. Moreover, sufficient conditions are obtained for a $q$-commuting triple of contractions with $\mathrm{\parallel }q\mathrm{\parallel }=1$ to admit a $q$-isometric dilation.