In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov–Sergeev solution to the Zamolodchikov tetrahedron equation is characterized as the transition matrix for a particular case of type A, which clarifies an algebraic origin of it. Our work is inspired by the recent developments connecting transition matrices for quantum non-super algebras with intertwiners of irreducible representations of quantum coordinate rings. We also discuss the crystal limit of transition matrices, which gives a super analog of transition maps of Lusztig’s parametrizations of the canonical basis.

在本文中，我们研究了在2级和3级情况下与所有可能的A型和B型Dynkin图相关的量子超级代数的幂等子代的PBW基的跃迁矩阵，并研究了与三维（3D）可积性的关系。我们通过类型A获得Zamolodchikov四面体方程的新解，通过类型B获得3D反射方程的新解，其中后者由Isaev和Kulish提出，是Cherednik反射方程的3D模拟。作为我们方法的副产品，Zamolodchikov四面体方程的Bazhanov-Sergeev解的特征是特定类型A情况的转移矩阵，从而阐明了其代数起源。我们的工作受到近期发展的启发，该发展将量子非超代数的跃迁矩阵与量子坐标环的不可约表示的缠结联系在一起。我们还讨论了跃迁矩阵的晶体极限，它给出了典型的Lusztig参数化跃迁图的超级模拟。