?_? graded discrete Lax pairs and integrable difference equations

We introduce a class of Z_N graded discrete Lax pairs, with N×N matrices, linear in the

spectral parameter. We give a classification scheme for such Lax pairs and the associated

discrete integrable systems. We present two potential forms and completely classify the

generic case. Many well known examples belong to our scheme for N = 2, so many of our

systems may be regarded as generalisations of these. Even at N = 3, several new integrable

systems arise. Many of our equations are mutually compatible, so can be used together to

form “coloured” lattices.