τ-functions: evolution and determinant representations

We discuss generalized τ-function in fundamental representations and constrain evolution operators to be always group elements. We discuss implications of this postulate for the cases of Lie groups S L(N) and quantum groups S Lq(N). It turns out that the postulate needs to be slightly modified to take into account that no ''nilpotent subgroups'' exist in S Lq(N) for q ≠ 1. This has some definite and simple implications for q-determinant-like representations of quantum τ-functions. (orig.)