  This paper begins investigation of the concept of ``generalized
$\tau$-function'', defined as a generating function of all the matrix elements
of a group element $g \in G$ in a given highest-weight representation of a
universal enveloping algebra ${\cal G}$. In the generic situation, the
time-variables correspond to the elements of maximal nilpotent subalgebras
rather than Cartanian elements. Moreover, in the case of quantum groups such
$\tau$-``functions'' are not $c$-numbers but take their values in
non-commutative algebras (of functions on the quantum group $G$). Despite all
these differences from the particular case of conventional $\tau$-functions of
integrable (KP and Toda lattice) hierarchies (which arise when $G$ is a
Kac-Moody (1-loop) algebra of level $k=1$), these generic $\tau$-functions also
satisfy bilinear Hirota-like equations, which can be deduced from manipulations
with intertwining operators. The most important applications of the formalism
should be to $k>1$ Kac-Moody and multi-loop algebras, but this paper contains
only illustrative calculations for the simplest case of ordinary (0-loop)
algebra $SL(2)$ and its quantum counterpart $SL_q(2)$, as well as for the
system of fundamental representations of $SL(n)$.
