Heisenberg Double and Pentagon Relation

TL;DR

This paper demonstrates that the Heisenberg double possesses a canonical element satisfying the pentagon relation, linking it to algebraic structures like the Drinfeld double and solutions to the Yang-Baxter equation.

Contribution

It establishes a canonical element in the Heisenberg double that satisfies the pentagon relation and connects it to the Drinfeld double and Yang-Baxter solutions.

Findings

Heisenberg double has a canonical element satisfying the pentagon relation

Drinfeld double can be embedded in the tensor square of the Heisenberg double

Solutions to the Yang-Baxter relation can be derived from pentagon solutions

Abstract

It is shown that the Heisenberg double has a canonical element, satisfying the pentagon relation. From a given invertible constant solution to the pentagon relation one can restore the structure of the underlying algebras. Drinfeld double can be realized as a subalgebra in the tensor square of the Heisenberg double. This enables one to write down solutions to the Yang-Baxter relation in terms of solutions to the pentagon relation.