Abstract

We define quantum determinants in quantum matrix algebras related to pairs of compatible braidings. We establish relations between these determinants and the so-called column and row determinants, which are often used in the theory of integrable systems. We also generalize the quantum integrable spin systems using generalized Yangians related to pairs of compatible braidings. We demonstrate that such quantum integrable spin systems are not uniquely determined by the “quantum coordinate ring” of the basic space \(V\). For example, the “quantum plane” \(xy=qyx\) yields two different integrable systems: rational and trigonometric.

中文翻译：

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量子矩阵代数和可积系统中的行列式

摘要

我们在与兼容编织对相关的量子矩阵代数中定义量子行列式。我们建立了这些行列式和所谓的列行列式之间的关系，这些行列式在可积系统理论中经常使用。我们还使用与兼容编织对相关的广义扬子来概括量子可积自旋系统。我们证明了这种量子可积自旋系统不是由基本空间\(V\)的“量子坐标环”唯一确定的。例如，“量子平面” \(xy=qyx\)产生两个不同的可积系统：有理系统和三角系统。