We give a description of the centralizer algebras for tensor powers of spin objects in the pre-modular categories $SO(N)_2$ (for $N$ odd) and $O(N)_2$ (for $N$ even) in terms of quantum $(n-1)$-tori, via non-standard deformations of $U\mathfrak{so}_N$. As a consequence we show that the corresponding braid group representations are Gaussian representations, the images of which are finite groups. This verifies special cases of a conjecture that braid group representations coming from weakly integral braided fusion categories have finite image.

中文翻译：

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SO$(N)_2$ 编织群表示是高斯的

我们描述了预模类别 $SO(N)_2$（对于 $N$ 奇数）和 $O(N)_2$（对于 $N$ 偶数）中自旋对象张量幂的中心化代数量子 $(n-1)$-tori 的术语，通过 $U\mathfrak{so}_N$ 的非标准变形。因此，我们表明相应的辫子群表示是高斯表示，其图像是有限群。这验证了来自弱积分编织融合类别的编织群表示具有有限图像的猜想的特殊情况。