The Q-deformed SUC, BUC hierarchies and the multi-component generalizations

Abstract

In this paper, we construct two kinds of [Formula: see text]-differential integrable hierarchies. The first one is the [Formula: see text]-universal character of B-type (BUC) and lattice [Formula: see text]-BUC hierarchies expressed by [Formula: see text]-shift operators. It is also shown that the lattice [Formula: see text]-BUC hierarchy reduces to the [Formula: see text]-BUC hierarchy with certain conditions. The other one is the [Formula: see text]-deformed generalized symplectic Schur functions and the [Formula: see text]-deformed generalized Q-functions defined by [Formula: see text]-derivative operators. By means of the [Formula: see text]-deformed vertex operators, the [Formula: see text]-deformed symplectic universal character (SUC) and BUC hierarchies have been presented. Moreover, it is proved that [Formula: see text]-deformed generalized symplectic Schur functions and generalized Q-functions are solutions of the [Formula: see text]-deformed SUC and BUC hierarchies, respectively. Meanwhile, by using multi-component tau functions, we establish the multi-component [Formula: see text]-deformed SUC and BUC hierarchies. In addition, based upon the discrete tau functions, the multi-component modified [Formula: see text]-deformed SUC and BUC hierarchies have been discussed.