  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.
