Hirota bilinear discrete equation and associated commutator identity

It is demonstrated that not only integrable pde's but also difference equations are in correspondence with some commutator identities on associative algebra. Specific commutator identity is suggested and it is shown that by means of a dresing procedure the Hirota bilinear discrete equation (HBDE) as well as its Lax pair can be derived. Direct and inverse spectral problems and Backlund transformation for HBDE are considered.