Additional Symmetries Of The KP Hierarchy And Its Q-deformation

In this paper we study the additional symmetries and the associated problem of the KP hierarchy and q-KP hierarchy. The Virasoro constraints on the r function and the explicit forms of the Virasoro generators of the BKP hierarchy are obtained. After constructing the additional symmetries of the CKP hierarchy, we show that the additional symmetry flows form a infinite dimension algebra W_1+∝^C, and then we derive the string equation and the Virasoro constraints on the Lax operator L of the CKP hierarchy. Based on the special additional symmetry flows (?)_t1,-(n-1)^*, the string equation of the q-KP hierarchy is given first, and then the Virasoro constraints of the 2-reduced q-KP hierarchy(i.e. q-KdV hierarchy) are also calculated explicitly.