In this paper, we study two deformation procedures for quantum groups: deformations by twists, that we call “comultiplication twisting”, as they modify the coalgebra structure, while keeping the algebra one — and deformations by 2-cocycle, that we call “multiplication twisting”, as they deform the algebra structure, but save the coalgebra one. We deal with quantized universal enveloping algebras (in short QUEAs), for which we accordingly consider those arising from twisted deformations (in short TwQUEAs) and those arising from 2-cocycle deformations, usually called multiparameter QUEAs (in short MpQUEAs). Up to technicalities, we show that the two deformation methods are equivalent, in that they eventually provide isomorphic outputs, which are deformations (of either kinds) of the “canonical”, well-known one-parameter QUEA by Jimbo and Lusztig. It follows that the two notions of TwQUEAs and of MpQUEAs — which, in Hopf algebra theoretical terms are naturally dual to each other — actually coincide; thus, that there exists in fact only one type of “pluriparametric deformation” for QUEAs. In particular, the link between the realization of any such QUEA as a MpQUEA and that as a TwQUEA is just a (very simple, and rather explicit) change of presentation.

中文翻译：

---

量子群的扭曲变形与共循环变形

在本文中，我们研究了量子群的两种变形过程：扭曲变形，我们称之为“共乘扭曲”，因为他们修改了代数结构，同时保持代数为一——以及变形2-cocycle，我们称之为“乘法扭曲”，因为它们使代数结构变形，但保留了余代数。我们处理量化的通用包络代数（简称 QUEA），为此我们相应地考虑那些由扭曲变形（简称 TwQUEA）产生的代数和那些由2-cocycle 变形，通常称为多参数 QUEA（简称 MpQUEA）。就技术细节而言，我们证明这两种变形方法是等效的，因为它们最终提供同构输出，即 Jimbo 和 Lusztig 著名的“规范”单参数 QUEA 的变形（任何一种）。因此，TwQUEA 和 MpQUEA 的两个概念——在 Hopf 代数理论术语中自然是相互对偶的——实际上是一致的。因此，对于 QUEA，实际上只存在一种类型的“多参数变形”。特别是，任何这样的 QUEA（如 MpQUEA）与 TwQUEA 的实现之间的联系只是（非常简单且相当明确的）表示的更改。