Cocommutative q-cycle coalgebra structures on the dual of the truncated polynomial algebra

TL;DR

This paper classifies certain algebraic structures called q-cycle coalgebras on the dual of truncated polynomial algebras, introducing a family called Standard Cycle Coalgebras and exploring their relation to the braid equation.

Contribution

It provides a partial classification of q-cycle coalgebra structures on the dual of truncated polynomial algebras and introduces Standard Cycle Coalgebras parameterized by free parameters.

Findings

Introduction of Standard Cycle Coalgebras parameterized by {p_1,...,p_{n-1}}

Verification of compatibility with the braid equation via differential operators

Potential connection between these operators and Yang's operators in quantum field theory

Abstract

In order to construct solutions of the braid equation we consider bijective left non-degenerate set-theoretic type solutions, which correspond to regular q-cycle coalgebras. We obtain a partial classification of the different q-cycle coalgebra structures on the dual coalgebra of $K[y]/\langle y^n\rangle$, the truncated polynomial algebra. We obtain an interesting family of involutive q-cycle coalgebras which we call Standard Cycle Coalgebras. They are parameterized by free parameters $\{p_1,...,p_{n-1}\}$ and in order to verify that they are compatible with the braid equation, we have to verify that certain differential operators $\partial^j$ on formal power series in two variables $K[[x, y]]$ satisfy the condition $(\partial^j G)_i = (\partial^i G)_j$ for all i, j, where $G$ is a formal power series associated to the given q-cycle coalgebra. It would be interesting to find out the relation of these operators with the operators given by Yang in the context with 2-dimensional quantum field theories, which was one of the origins of the Yang-Baxter equation.