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 ℒ and Orlov–Schulman function ℳ of the hierarchies. Meanwhile, the additional symmetry will be studied with the infinite flows of vi and τ 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.

中文翻译：

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无色散扩展非交换 Gelfand-Dickey 层次结构的附加对称性

本文旨在研究未扩展和扩展、交换和非交换无色散 Gelfand-Dickey (dGD) 层次结构的额外对称性。类似于 Gelfand-Dickey (GD) 层次结构的 Lax 形式，我们构造函数ℒ和 Orlov–Schulman 函数ℳ的层次结构。同时，将研究附加对称性与无限流v一世和τdGD 层次结构的函数，可以发现只有一部分附加流可以在具有相应字符串方程的 GD 约束下存活。此外，我们关注具有量子环面代数结构的无色散扩展 Gelfand-Dickey (dEGD) 层次结构的额外对称性，并详细显示了流动。研究了无色散非交换Gelfand-Dickey（dNCGD）层次和无色散扩展非交换Gelfand-Dickey（dENCGD）层次的附加对称性。