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Classification of $mathfrak{sl}_3$ relations in the Witt group of nondegenerate braided fusion categories

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 Added by Andrew Schopieray
 Publication date 2016
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and research's language is English




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The Witt group of nondegenerate braided fusion categories $mathcal{W}$ contains a subgroup $mathcal{W}_text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $mathcal{C}(mathfrak{g},k)$ consisting of highest weight integerable $hat{g}$-modules of level $k$ where $hat{mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[mathcal{C}(mathfrak{sl}_2,k)]$, $kgeq1$ have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[mathcal{C}(mathfrak{sl}_3,k)]$, $kgeq1$ with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras $mathfrak{g}$ and positive integer levels $k$.



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177 - Andrew Schopieray 2021
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generalize to braided fusion categories with the fusion rules of the representation categories of extraspecial $p$-groups for any prime $p$, which possess a minimal modular extension only if they arise as the adjoint subcategory of a twisted double of an extraspecial $p$-group.
We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has F if the associated braid group representations factor over a finite group, and suggest that categories of integral Frobenius-Perron dimension are precisely those with property F.
85 - E. Feigin 2006
In this paper we study an approximation of tensor product of irreducible integrable $hat{mathfrak{sl}_2}$ representations by infinite fusion products. This gives an approximation of the corresponding coset theories. As an application we represent characters of spaces of these theories as limits of certain restricted Kostka polynomials. This leads to the bosonic (which is known) and fermionic (which is new) formulas for the $hat{mathfrak{sl}_2}$ branching functions.
137 - Victor Ostrik , Zhiqiang Yu 2021
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