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 $u= (u_{ i,j,k,l})$ is an $nm \times nm$ 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 $u$.
Categories: math.OA