Additional symmetries of the dispersionless extended noncommutative Gelfand–Dickey hierarchy

This paper aims at additional symmetries of the unextended and extended, commutative and noncommutative dispersionless Gelfand–Dickey (dGD) hierarchies. Being similar to the Lax formalism of the Gelfand–Dickey (GD) hierarchy, we construct the function [Formula: see text] and Orlov–Schulman function [Formula: see text] of the hierarchies. Meanwhile, the additional symmetry will be studied with the infinite flows of [Formula: see text] and [Formula: see text] function of the dGD hierarchy and one can find that only a part of additional flows can survive under the GD constraints with the corresponding string equation. Furthermore, we pay attention to the additional symmetries of the dispersionless extended Gelfand–Dickey (dEGD) hierarchy which has a quantum torus algebraic structure and show the flows in detail. The additional symmetry of dispersionless noncommutative Gelfand–Dickey (dNCGD) hierarchy and dispersionless extended noncommutative Gelfand–Dickey (dENCGD) hierarchy are studied.

Additional symmetries of the Two-Boson hierarchy and the multi-component Two-Boson hierarchy

In this paper, we firstly recall some basic facts on the Two-Boson hierarchy. Then, introducing some time variables that consist of a non-abelian Lie algebra. Next, we construct additional symmetries for the Two-Boson hierarchy with the aid of the Orlov–Schulman operator, which depend on the time variables and dressing operator. In addition, we give the additional flow equations of the Two-Boson hierarchy as a simple example, and prove that the additional flows are symmetries of the Two-Boson hierarchy. In this way, an isomorphism between the additional symmetries of the Two-Boson hierarchy and the [Formula: see text] algebra is constructed. Finally, the multi-component Two-Boson hierarchy can be defined, and we consider the additional symmetries for the multi-component Two-Boson hierarchy with the method of Dickey, Orlov, and Shulman.

Additional symmetries for the supersymmetric Gelfand–Dickey hierarchy

In this paper, we first define a supersymmetric Gelfand–Dickey hierarchy using the Sato equation. Then we introduce some time variables and derivations involving those that from a non-abelian Lie superalgebra. Next, we construct additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the aid of the Orlov–Schulman operators which involve the above time variables and dressing operators. We give the additional flow equations of the supersymmetric Gelfand–Dickey hierarchy, and prove that the additional flows are symmetries of the supersymmetric Gelfand–Dickey hierarchy. In this way, an isomorphism between the additional symmetries of the supersymmetric Gelfand–Dickey hierarchy and the super [Formula: see text] algebra is constructed. Finally, the [Formula: see text] type supersymmetric Gelfand–Dickey hierarchy is constructed and we consider the additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the [Formula: see text] type condition which is compatible with the flows of the [Formula: see text] type supersymmetric Gelfand–Dickey hierarchy.