Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators $L_{e_1},…, L_{e_n}, L_{f_1},…,L_{f_m}$ subject to the unitary commutation relations of the form $$L_{e_i}{L}_{f_j}=\sum _{k,l}{u}_{i,j,k,l}{L}_{f_l}{L}_{e_k}$$ where $u= (u_{i,j,k,l})$isan$nm \times nm$ unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix $u$.