Tetrahedron and 3D Reflection Equation from PBW Bases of the Nilpotent Subalgebra of Quantum Superalgebras

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...

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Bibliographic Details
Published in:Communications in mathematical physics 2021-10, Vol.387 (1), p.481-550
Main Author: Yoneyama, Akihito
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Mathematical analysis
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Rings (mathematics)
Tetrahedra
Theoretical
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Summary: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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-021-04098-8