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arXiv · 2107.08340

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

Jorge Guccione·Juan José Guccione·Christian Valqui
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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.

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Jorge Guccione, Juan José Guccione, Christian Valqui. 2021-07-18. Cocommutative q-cycle coalgebra structures on the dual of the truncated polynomial algebra. https://arxiv.org/abs/2107.08340

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