Tetrahedron and 3D Reflection Equation from PBW Bases of the Nilpotent Subalgebra of Quantum Superalgebras
Tetrahedron and 3D Reflection Equation from PBW Bases of the Nilpotent Subalgebra of Quantum Superalgebras
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DOI：
10.1007/s00220-021-04098-8
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发表时间：
2020-12
影响因子：
2.4
通讯作者：
Akihito Yoneyama
中科院分区：
文献类型：
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作者：
Akihito Yoneyama
在本文中，我们研究了在 2 阶和 3 阶情况下与 A 型和 B 型所有可能的 Dynkin 图相关的量子超代数的幂零子代数的 PBW 基的转移矩阵，并检验了与三维 (3D) 可积性的关系。我们通过 A 型获得了 Zamolodchikov 四面体方程的新解，通过 B 型获得了 3D 反射方程的新解，其中后者方程是由 Isaev 和 Kulish 提出的，作为 Cherednik 反射方程的 3D 模拟。作为我们方法的副产品，Zamoodchikov 四面体方程的 Bazhanov-Sergeev 解被描述为 A 型特定情况的过渡矩阵，这澄清了它的代数起源。我们的工作受到最近将量子非超代数的过渡矩阵与量子坐标环的不可约表示的交织器连接起来的发展的启发。我们还讨论了过渡矩阵的晶体极限，它给出了 Lusztig 规范基参数化过渡图的超级模拟。
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.