On the classification of q-algebras

The problem is the classification of the ideals of 'free differential algeb ras', or the associated quotient algebras, the q-algebras; being finitely g enerated, unital C-algebras with homogeneous relations and a q-differential structure. This family of algebras includes the quantum groups, or at leas t those that are based on simple (super) Lie or Kac-Moody algebras. Their c lassification would encompass the so far incompleted classification of quan tized (super) Kac-Moody algebras and of the (super) Kac-Moody algebras them selves. These can be defined as singular limits of q-algebras, and it is ev ident that to deal with the q-algebras in their full generality is more rat ional than the examination of each singular limit separately. This is not j ust because quantization unifies algebras and superalgebras, but also becau se the points 'q=1' and 'q=-1' are the most singular points in parameter sp ace. In this Letter, one of two major hurdles in this classification progra m has been overcome. Fix a set of integers n(1),...,n, and consider the spa ce B-Q of homogeneous polynomials of degree n(1) in the generator e(1), and so on. Assume that there are no constants among the polynomials of lower d egree, in any one of the generators; in this case all constants in the spac e B-Q have been classified. The task that remains, the more formidable one, is to remove the stipulation that there are no constants of lower degree.