Tetrahedron and 3D reflection equations from quantized algebra of functions

TL;DR

This paper constructs new solutions to the 3D reflection equation using quantized algebra of functions, extending existing theories to symplectic groups and exploring polynomial, combinatorial, and birational structures.

Contribution

It extends quantized function algebra methods to Sp_{2n} and provides the first solutions to the 3D reflection equation, including polynomial and combinatorial aspects.

Findings

Constructed intertwiner K for A_q(Sp_{2n})

Derived solutions to the 3D reflection equation

Identified polynomial, combinatorial, and birational structures

Abstract

Soibelman's theory of quantized function algebra A_q(SL_n) provides a representation theoretical scheme to construct a solution of the Zamolodchikov tetrahedron equation. We extend this idea originally due to Kapranov and Voevodsky to A_q(Sp_{2n}) and obtain the intertwiner K corresponding to the quartic Coxeter relation. Together with the previously known 3-dimensional (3D) R matrix, the K yields the first ever solution to the 3D analogue of the reflection equation proposed by Isaev and Kulish. It is shown that matrix elements of R and K are polynomials in q and that there are combinatorial and birational counterparts for R and K. The combinatorial ones arise either at q=0 or by tropicalization of the birational ones. A conjectural description for the type B and F_4 cases is also given.