On the classification of q-algebras

Abstract

The problem is the classification of the ideals of ``free differential algebras", or the associated quotient algebras, the q-algebras; being finitely generated, unital C-algebras with homogeneous relations and a q-differential structure. This family of algebras includes the quantum groups, or at least those that are based on simple (super) Lie or Kac-Moody algebras. Their classification would encompass the so far incompleted classification of quantized (super) Kac-Moody algebras and of the (super) Kac-Moody algebras themselves. These can be defined as singular limits of q-algebras, and it is evident that to deal with the q-algebras in their full generality is more rational than the examination of each singular limit separately. This is not just because quantization unifies algebras and superalgebras, but also because the points "q = 1" and "q = -1" are the most singular points in parameter space. In this paper one of two major hurdles in this classification program has been overcome. Fix a set of integers n_1,...,n_k, and consider the space 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 degree, in any one of the generators; in this case all constants in the space 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.