Wold-type decomposition for $\mathcal{U}_n$-twisted contractions

Let $n>1$, and $\{U_{ij}\}$ for $1 \leq i < j \leq n$ be $\binom{n}{2}$ commuting unitaries on a Hilbert space $\mathcal{H}$ such that $U_{ji}:=U^*_{ij}$. An $n$-tuple of contractions $(T_1, \dots, T_n)$ on $\mathcal{H}$ is called $\mathcal{U}_n$-twisted contraction with respect to a twist $\{U_{ij}\}_{i<j}$ if $T_1, \dots, T_n$ satisfy $$T_iT_j=U_{ij}T_jT_i; \hspace{0.5cm} \hspace{1cm} T_i^*T_j= U^*_{ij}T_jT_i^* \hspace{0.5cm} \mbox{and} \hspace{0.5cm} T_kU_{ij} =U_{ij}T_k$$ for all $i,j,k=1, \dots, n$ and $i \neq j$. We obtain a recipe to calculate the orthogonal spaces of the Wold-type decomposition for $\mathcal{U}_n$-twisted contractions on Hilbert spaces. As a by-product, a new proof as well as complete structure for $\mathcal{U}_2$-twisted (or pair of doubly twisted) and $\mathcal{U}_n$-twisted isometries have been established.