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## Algebraic properties of Universal Quantum Semigroupoids

### Fabio Calderón

#### Universidad Nacional de Colombia, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak8a3e432ca35d5dc7b80aaaa6ceef7b0f').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy8a3e432ca35d5dc7b80aaaa6ceef7b0f = 'f&#97;c&#97;ld&#101;r&#111;nm' + '&#64;'; addy8a3e432ca35d5dc7b80aaaa6ceef7b0f = addy8a3e432ca35d5dc7b80aaaa6ceef7b0f + '&#117;n&#97;l' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;'; var addy_text8a3e432ca35d5dc7b80aaaa6ceef7b0f = 'f&#97;c&#97;ld&#101;r&#111;nm' + '&#64;' + '&#117;n&#97;l' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;';document.getElementById('cloak8a3e432ca35d5dc7b80aaaa6ceef7b0f').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy8a3e432ca35d5dc7b80aaaa6ceef7b0f + '\'>'+addy_text8a3e432ca35d5dc7b80aaaa6ceef7b0f+'<\/a>';

The study of quantum symmetries over a non-connected graded algebra $A$ leads to the concept of (co)actions of weak bialgebras and, in particular, the Universal Quantum Semigroupoids (UQSGs). A remarkable set-up in which these UQSGs arise is when the algebra $A=\Bbbk Q$ is a path algebra over a finite quiver $Q$.

Last year, H. Huang, C. Walton, E. Wicks and R. Won proved two relevant results: 1. the associated UQSG of $A=\Bbbk Q$ is isomorphic to the Hayashi's face algebra $\mathfrak{H}(Q)$ attached to $Q$, and 2. when $Q$ is an extended Dynkin quiver the associated UQSG of the preprojective algebra $A=\Pi_Q$ attached to $Q$ is isomorphic to a certain quotient of $\mathfrak{H}(Q)$.

In joint work with Chelsea Walton, we study some ring-theoretic and homological properties of $\mathfrak{H}(Q)$ using the previous results. We will provide the motivation behind this research, the results already obtained and the following steps of our study.

Joint work with Chelsea Walton (Rice University, TX, USA).