Small modules over quantum complete intersections in two variables

We describe the left regular module of a quantum complete intersection A(q,n(1),...,n(t)) by the property that it is the unique finite-dimensional indecomposable left A(q,n(1),...,n(t))-module of Loewy length n-ary sumation Sigma(1 <= i <= t) n(i) - t + 1. Using a reduction to A(q, 4, 4)-modules, we classify the 4-dimensional indecomposable left modules over quantum complete intersection A(q,m,n) in two variables, by explicitly giving their diagram presentations. Together with the existed work on indecomposable A(q,m,n)-modules of dimension <= 3, we then know all the indecomposable A(q,m,n)-modules of dimension <= 4.