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## Zesting of Fusion Categories from quantum groups of Type $A$

### Hector Giovanny Mora Diaz

#### Universidad de los Andes, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloaka857a215c07673c1f1500255a0a27590').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addya857a215c07673c1f1500255a0a27590 = 'hg.m&#111;r&#97;' + '&#64;'; addya857a215c07673c1f1500255a0a27590 = addya857a215c07673c1f1500255a0a27590 + '&#117;n&#105;&#97;nd&#101;s' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;'; var addy_texta857a215c07673c1f1500255a0a27590 = 'hg.m&#111;r&#97;' + '&#64;' + '&#117;n&#105;&#97;nd&#101;s' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;';document.getElementById('cloaka857a215c07673c1f1500255a0a27590').innerHTML += '<a ' + path + '\'' + prefix + ':' + addya857a215c07673c1f1500255a0a27590 + '\'>'+addy_texta857a215c07673c1f1500255a0a27590+'<\/a>';

Zesting is a construction of a new braided fusion category from a given braided fusion category by defining a new tensor product.

If $\mathcal{C}=\bigoplus_{a \in A} \mathcal{C}_a$ is an $A$-graded fusion category where $A$ is an abelian group, we define a new tensor product $X_a\stackrel{\lambda}{\otimes} Y_b:=(X_a\,\otimes\, Y_b)\otimes\,\lambda(a,b)$, where $X_a\in\mathcal{C}_a$ and $Y_b\in\mathcal{C}_b$ are simple objects in their corresponding graded components and $\lambda(a,b)\in\mathcal{C}_e$ is an invertible object in the trivial component.

In this work, I present a total description of the Braided Zestings from the modular category $SU(N)_k$ obtained from the quantum group $U_q (\mathfrak{sl}_n)$ extending the work on [2] by giving explicit calculations for finding of all the braided zestings obtained from $SU(N)_k$. \textbf{ References:} [1] C. {Delaney} and C. {Galindo} and J. {Plavnik} and E. {Rowell} and Q. {Zhang}, \textit{Braided Zestings and Aplications}. Communication in Mathematical Physics. [2] H. Mora, \textit{Zestings de categorías modulares construidas a partir de grupos cu\'anticos de tipo A}. Masther Thesis. Universidad de los Andes, Colombia.