Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras

TL;DR

This paper explores the algebraic structures of quantum superalgebras, deriving new solutions to the Zamolodchikov tetrahedron and 3D reflection equations through PBW bases, and connects these to integrability and crystal bases.

Contribution

It introduces novel solutions to 3D integrability equations using PBW bases of quantum superalgebras, linking algebraic and geometric aspects of quantum groups.

Findings

New solutions to Zamolodchikov tetrahedron and 3D reflection equations.

Characterization of Bazhanov-Sergeev solution as a transition matrix.

Discussion of crystal limit leading to super analogs of Lusztig's parametrizations.

Abstract

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.