Tetrahedron equation, Weyl group, and quantum dilogarithm

TL;DR

This paper constructs solutions to the tetrahedron equation utilizing quantum groups, Weyl group elements, and quantum dilogarithm functions, advancing the understanding of quantum algebraic structures.

Contribution

It introduces a novel family of solutions to the tetrahedron equation derived from the RTT presentation of a two-parameter quantized algebra, incorporating Weyl group and quantum dilogarithm techniques.

Findings

Derived explicit solutions to the tetrahedron equation.

Connected quantum dilogarithm with quantum group structures.

Highlighted the role of Weyl group elements in solutions.

Abstract

We derive a family of solutions to the tetrahedron equation using the RTT presentation of a two parametric quantized algebra of regular functions on an upper triangular subgroup of GL(n). The key ingredients of the construction are the longest element of the Weyl group, the quantum dilogarithm function, and central elements of the quantized division algebra of rational functions on the subgroup in question.