Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators Le1,…,Len,Lf1,…,LfmLe1,…,Len,Lf1,…,Lfm subject to the unitary commutation relations of the formLeiLfj=∑k,lui,j,k,lLflLek where u=(ui,j,k,l)u=(ui,j,k,l) is an nm×nmnm×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.