Symmetries of the multicomponent $q$-KP hierarchy on a Grassmannian

Based on the study of quantum calculus, we construct a multicomponent $q$-KP hierarchy and its additional symmetries. The additional symmetries form a multifold $W_{1+\infty}$ algebra and the generating operator of the additional symmetries can be shown to have a concise form in terms of wave functions. Furthermore, the string equation and the action of additional symmetries of the multicomponent $q$-KP hierarchy on the Grassmannian are considered. After quantization, we derive the corresponding quantum torus symmetry, whose flows constitute an interesting multifold quantum torus type Lie algebra.