This is a report on a joint work with Jason Green. We define a Tannakian radical of a braided fusion category C as the intersection of its maximal Tannakian subcategories. The localization of C corresponding to the Tannakian radical (called the mantle) has a canonical central extension which is a complete invariant of C. We study the properties of this invariant and discuss how it can be used in the classification of braided fusion categories.
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