## View abstract

### Session S16 - Quantum symmetries

Monday, July 19, 17:45 ~ 18:10 UTC-3

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

### Christian Valqui

#### Pontificia Universidad Católica del Perú, Perú   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak6a5cc375d03882954fca6644e072e354').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy6a5cc375d03882954fca6644e072e354 = 'cv&#97;lq&#117;&#105;' + '&#64;'; addy6a5cc375d03882954fca6644e072e354 = addy6a5cc375d03882954fca6644e072e354 + 'p&#117;cp' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'p&#101;'; var addy_text6a5cc375d03882954fca6644e072e354 = 'cv&#97;lq&#117;&#105;' + '&#64;' + 'p&#117;cp' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'p&#101;';document.getElementById('cloak6a5cc375d03882954fca6644e072e354').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy6a5cc375d03882954fca6644e072e354 + '\'>'+addy_text6a5cc375d03882954fca6644e072e354+'<\/a>';

In order to construct solutions of the braid equation we consider biijective 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.

Joint work with Juan José Guccione (Universidad de Buenos Aires) and Jorge Guccione (Universidad de Buenos Aires).