It is usually assumed, that some version of the central extended $GL(\infty)$ group acts transitively on the space of tau-function of the KP integrable hierarchy. It means that any tau-function can be obtained by the action of some group element on any other tau-function (say, a trivial one $\tau=1$).
1) How should I define the corresponding $GL(\infty)$? For example V. Kac in his book "Infinite dimensional Lie algebras" defines corresponding algebra as $$ \tilde{gl}_\infty:=\left\{(a_{ij})_{i,j\in\mathbb{Z}})|\text{ all but a finite number of $a_{ij}$ with $i\geq j$ are $0$}\right\}. $$ However, it is clear that this definition is too restrictive - in the applications we need the more general algebra elements with an infinite number of the non-zero matrix elements. In the book of M. Jimbo, T. Miwa and E. Date, "Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras" they consider the finite-band matrices, which is also too restrictive. The question is: how should we define the $gl_\infty$ algebra to describe all possible tau-functions?
2) Assume we have a proper definition of $gl_\infty$. How can we prove that corresponding group acts on the space of the tau-functions transitively? In particular, if I have a tau-function of the KP hierarchy, can I restore a corresponding group element (which is not unique, of course)?
