We revisit the derivation of Knizhnik–Zamolodchikov equations in the case of non-semisimple categories of modules of a superalgebra in the case of the generic affine level and representations parameters. A proof of existence of asymptotic solutions and their properties for the superalgebra \({\mathfrak {gl}}(1|1)\) gives a basis for the proof of existence of associator which satisfy braided tensor categories requirements. Braided tensor category structure of \(U_h({\mathfrak {gl}}(1|1))\) quantum algebra is calculated, and the tensor product ring is shown to be isomorphic to \({\mathfrak {gl}}(1|1)\) ring, for the same generic relations between the level and parameters of modules. We review the proof of Drinfeld–Kohno theorem for non-semisimple category of modules suggested by Geer (Adv Math 207:1–38, 2006) and show that it remains valid for the superalgebra \({\mathfrak {gl}}(1|1)\). Examples of logarithmic solutions of KZ equations are also presented.

在通用仿射级和表示参数的情况下，我们在超级代数的模块的非半简单类别的情况下，重新研究Knizhnik–Zamolodchikov方程的推导。证明超代数\（{\ mathfrak {gl}}（1 | 1）\）的渐近解及其性质，为满足编织张量类别要求的关联子的存在提供了依据。计算\（U_h（{\ mathfrak {gl}}（1 | 1））\）量子代数的编织张量类别结构，并且张量积环与\（{\ mathfrak {gl}}（ 1 | 1）\）环，用于模块的级别和参数之间的相同通用关系。我们回顾了Geer建议的非半简单模块类别的Drinfeld–Kohno定理的证明（高级数学207：1–38，2006），并表明它对于超代数\（{\ mathfrak {gl}}（1 | 1）\）。还给出了KZ方程的对数解的示例。