Super Grassmann hierarchies -- A multicomponent theory (Q1103790)

This paper is concerned with a supersymmetric extension of the finite dimensional version of the multicomponent Kadomtsev-Petviashvili (KP) hierarchy. The (one-component) supersymmetric KP hierarchy was first introduced by \textit{Yu. I. Manin} and \textit{A. O. Radul} [Commun. Math. Phys. 98, 65-77 (1985; Zbl 0607.35075)] and was studied extensively by Ueno and the author [Adv. Stud. Pure Math. 16, 373-426 (1988)]. It is a natural question how to extend these theories to the multicomponent case. For a Grassmann number \(\theta\), put \(\Theta =\partial_{\theta}+\theta \partial_ x\), and let \(W=\sum^{m}_{j=0}w_ j\Theta^{-j}\) be a monic super-microdifferential operator with matrix coefficients \(w_ j=(w_ j^{(\alpha \beta)})_{0\leq \alpha,\beta <r}\). Then the multicomponent SKP hierarchy is described by the Sato equations: \[ \Theta_{2n}^{(\alpha)}(W)=(-)^ n(B_{2n}^{(\alpha)}W- WE_{\alpha \alpha}\Theta^{2n}),\quad \Theta^{(\alpha)}_{2n- 1}(W)=(-)^{n+m}(B^{(\alpha)}_{2n-1}W-(-)^{\alpha}JWE_{\alpha \alpha}\Theta^{2n-1}), \] where \(\Theta_ k^{(\alpha)}\)'s are the super-time evolution operators and \(J=\sum^{r-1}_{\alpha =0}(- )^{\alpha}E_{\alpha \alpha}.\) The `\(\tau\)-function' is defined as the superdeterminant of certain super-Wronski matrix attached to a point of the super Grassmann manifold. The coefficients \(w_ j\) of W are represented explicitly as the quotients of the superdeterminants.