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Braided differential operators on quantum algebras

机译:量子代数上的编织微分算子

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摘要

We propose a general scheme of constructing braided differential algebras via algebras of "quantum exponentiated vector fields" and those of "quantum functions". We treat a reflection equation algebra as a quantum analog of the algebra of vector fields. The role of a quantum function algebra is played by a general quantum matrix algebra. As an example we mention the so-called RTT algebra of quantized functions on the linear matrix group GL(m). In this case our construction essentially coincides with the quantum differential algebra introduced by S. Woronowicz. If the role of a quantum function algebra is played by another copy of the reflection equation algebra we get two different braided differential algebras. One of them is defined via a quantum analog of (co)adjoint vector fields, the other algebra is defined via a quantum analog of right-invariant vector fields. We show that the former algebra can be identified with a subalgebra of the latter one. Also, we show that "quantum adjoint vector fields" can be restricted to the so-called "braided orbits" which are counterparts of generic GL(m)-orbits in gl*(m). Such braided orbits endowed with these restricted vector fields constitute a new class of braided differential algebras.
机译:我们提出了一种通过“量子指数矢量场”和“量子函数”的代数构造编织微分代数的一般方案。我们将反射方程代数视为矢量场代数的量子模拟。量子函数代数的作用是由一般的量子矩阵代数来实现的。作为示例,我们提到了线性矩阵组GL(m)上的所谓量化函数的RTT代数。在这种情况下,我们的构造与S. Woronowicz引入的量子微分代数基本重合。如果量子函数代数的作用是由反射方程代数的另一个副本扮演的,我们将得到两个不同的编织微分代数。其中一个是通过(共)伴随向量场的量子模拟定义的,另一个代数是通过右不变向量场的量子模拟定义的。我们证明前者代数可以与后者的子代数识别。同样,我们表明“量子伴随矢量场”可以被限制为所谓的“编织轨道”,它们是gl *(m)中通用GL(m)-轨道的对应物。赋予这些受限矢量场的这种编织轨道构成了一类新的编织微分代数。

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